Theorem

Let X be an object such that X is a transitive set. Let A be an object. Assume A∈X. Then we have A⊆X.

Background

The following background is necessary to understand this theorem.

(.) Let A and x be objects. Then x∈A is a proposition.

(.) Let A and B be objects. Then "A⊆B" is a proposition.

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

Background for Proof

The following background is necessary for the proof.

(.) Let A be an object. Let φ(a) be a proposition depending on a member a of A. Then "∀a∈A⋅φ(a)" is a proposition.

(.) Let A be an object. Let φ(a) be a proposition depending on a member a of A. Assume ∀a∈A⋅φ(a). Let a be a member of A. Then we know φ(a).

(.) Let A be an object. Then A is a transitive set is the proposition given by ∀a∈A⋅a⊆A.

Proof

We have ∀Y∈X⋅Y⊆X. Using this, we conclude A⊆X.

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