# Question 34722: Negation of a Finite Universal Statement [Return to catalogue page 174](https://ocme.evemisslab.com/en/questions/page/174/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r01-quantifier-foundations/#q-20261003-advance-20261004-r01-quantifier-0072) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0068-34722 · q-20261003-advance-20261004-r01-quantifier-0072 ## Original question and givens Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ¬(∀x∈U P(x)) true? Answer Yes or No. - G001 — Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ¬(∀x∈U P(x)) true? Answer Yes or No. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-quantifier](https://ocme.evemisslab.com/en/mko/mko-quantifier/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-quantifier-negated-universal, [U, operation, predicate]] After derived = [not_forall, no-invented-domain] Conditions: none Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact. Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · impliesBefore source roles = [quantifier ORIGINAL U, [9, 10, 12, 14]] After derived = [[9, 10, 12, 14]] Conditions: none THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol. Step identifiers and dependenciesRule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001 S003 · impliesBefore source roles = [{'kind': 'le', 'value': 12}] After derived = [Yes, validation only;0 is NOT added to U] Conditions: none The SOURCE validates predicateValue(0)EVEN for empty U:EXACT kind/value keys,SAFE value and a NONZERO multiple divisor.This is THIS protocol’s qualification,NOT adding0 to U or a universal theorem about divisibility by0. Step identifiers and dependenciesRule: Quantifier_SOURCE_predicate_validity_at0_EVEN_empty_U · Uses: G001, S002 S004 · impliesBefore source roles = [9, {'kind': 'le', 'value': 12}] After derived = [Yes] Conditions: none Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values. Step identifiers and dependenciesRule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S003 S005 · impliesBefore source roles = [10, {'kind': 'le', 'value': 12}] After derived = [Yes] Conditions: none Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values. Step identifiers and dependenciesRule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S004 S006 · impliesBefore source roles = [12, {'kind': 'le', 'value': 12}] After derived = [Yes] Conditions: none Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values. Step identifiers and dependenciesRule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S005 S007 · impliesBefore source roles = [14, {'kind': 'le', 'value': 12}] After derived = [No] Conditions: none Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values. Step identifiers and dependenciesRule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S006 S008 · impliesBefore source roles = [[9, 10, 12, 14], [[9, Yes], [10, Yes], [12, Yes], [14, No]], not_forall] After derived = [[9, 10, 12], [14], Yes] Conditions: none ∀ requires NO failures;∃ requires a passing member;¬∀ means a failing member;¬∃ means NO passing member.For empty U the four results are True/False/False/True,AFTER predicate qualification.No infinite-domain inference follows. Step identifiers and dependenciesRule: Quantifier_ALL_finite_truth_rows_negation_and_EMPTY_conventions · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S006 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S005 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S004 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. ## Result Yes The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 4ba80be1c88612083e73a2b6b887f6aae9518c9d4ae68223be42e73f6bfd2913 - candidate record SHA-256: b4f1b0772515e4ff625fa79384ae63a161cea5d1393fb2a34cd125d7c12289ed - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r01-quantifier-0072.json)