# Question 4347: Negated Universal and Existential Pair [Return to catalogue page 22](https://ocme.evemisslab.com/en/questions/page/22/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r07-quantifier-foundations/#q-20261008-advance-20261010-r07-quantifier-0047) 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. R0007-04347 · q-20261008-advance-20261010-r07-quantifier-0047 ## Original question and givens The explicitly finite integer domain is U=[-6, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention. - G001 — The explicitly finite integer domain is U=[-6, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention. ## 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 complete finite U=[-6, -5, -4, -3, -2, -1] After universal on empty domain=True;existential on empty domain=False Conditions: none Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain. Step identifiers and dependenciesRule: finite_quantifier_scope · Uses: G001 S002 · impliesBefore x=-6;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S001 S003 · impliesBefore x=-5;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S002 S004 · impliesBefore x=-4;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S003 S005 · impliesBefore x=-3;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S004 S006 · impliesBefore x=-2;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S005 S007 · impliesBefore x=-1;predicate=ge 0 After P(x)=No Conditions: none Evaluate the declared decidable integer predicate for each element,retaining both true and false cases. Step identifiers and dependenciesRule: finite_predicate_row · Uses: G001, S006 S008 · impliesBefore [[-6, No], [-5, No], [-4, No], [-3, No], [-2, No], [-1, No]] After T=[];F=[-6, -5, -4, -3, -2, -1];T∪F=U and T∩F=∅ Conditions: none The true and false parts are disjoint and cover the deduplicated domain;unlisted integers are not counterexamples. Step identifiers and dependenciesRule: complete_truth_partition · Uses: G001, S007 S009 · impliesBefore false-element count=6;true-element count=0 After (True, True) Conditions: none Not-all P means at least one false element,and not-any P means no true element;the empty domain therefore gives(False,True).Do not confuse the universal with its negation. Step identifiers and dependenciesRule: finite_quantifier_negations · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (True, True) 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: ec19db9a7fa743c1540f2560f10d0835dcccb7d6d509ae30ed88d15a387fb49f - candidate record SHA-256: 851bc79c3a105f0347142971439d82fd572a109ca291022450fda78ad0d0c55b - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r07-quantifier-0047.json)