Theorem

Let X be an object such that X is an ordinal. Let Y be an object such that Y is an ordinal. Then we have X∩Y is a transitive set.

Background

The following background is necessary to understand this theorem.

(.) Let A and B be objects. Then "A∩B" is an object.

(.) Let x be an object. Then "x is a transitive set" is a proposition.

(.) Let x be an object. Then "x is an ordinal" is a proposition.

Background for Proof

The following background is necessary for the proof.

(.) Let φ and ψ be propositions. Then φ∧ψ is a proposition.

(.) Let φ and ψ be propositions. Assume φ∧ψ. Then we know φ.

(.) Let X be an object such that X is a transitive set. Let Y be an object such that Y is a transitive set. Then we know X∩Y is a transitive set.

(.) Let x be an object. Then "x is well-ordered by membership" is a proposition.

(.) Let x be an object. Then x is an ordinal is the proposition given by "x is a transitive set"∧"x is well-ordered by membership".

Proof

Trivial.

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