# Question 5693: Original Membership and Set Insertion

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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.

R0010-05693 · q-20261008-advance-20261012-r04-set-membership-0043



## Original question and givens

Finite integer set A=[-17, 0], x=-17. Give (original truth of x∈A, sorted deduplicated A∪{x}).


- G001 — Finite integer set A=[-17, 0], x=-17. Give (original truth of x∈A, sorted deduplicated A∪{x}).





## 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-set-membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/)





## Step-by-step derivation

S001 · impliesBefore

declared finite type or bounded integer universe

After

membership truth is distinct from eligibility to ask the question

Conditions: none

Being outside the type is not the same as false membership.Check only the declared finite type or integer set,not all set theory or unrestricted comprehension.

Step identifiers and dependenciesRule: finite_membership_scope · Uses: G001





S002 · impliesBefore

original A=[-17, 0];x=-17

After

original membership=True

Conditions: none

Read original membership before changing the set;do not use the resulting set to overwrite the original judgment.

Step identifiers and dependenciesRule: original_set_membership_before_change · Uses: G001, S001





S003 · impliesBefore

operation=insert x=-17;original A=[-17, 0]

After

[-17, 0]

Conditions: none

Inserting an existing element adds no duplicate,and deleting an absent element leaves the set unchanged.Sort the result numerically while retaining the original source.

Step identifiers and dependenciesRule: finite_set_membership_update · Uses: G001, S002









## Reverse checks


- 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, [-17, 0])

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: e175fea53baa18eb025ad72efc03d6e5155bd2b7b28d708090020d28b5ab4a23
- candidate record SHA-256: dc7eecbc196af5e7320de8af8db6652acbf449c45c98c23f28b977e52094264d
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r04-set-membership-0043.json)