Theorem

Let A be an object. Let X and Y be members of ℘(A). Let a be a member of A. Assume X∈℘(Y) and a∈X. Then we have a∈Y.

Background

The following background is necessary to understand this theorem.

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

(.) Let A be an object. Then ℘(A) is an object.

Background for Proof

The following background is necessary for the proof.

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

(.) Let A, B and x be objects. Assume A⊆B and x∈A. Then we know x∈B.

(.) Let A and B be objects. Assume B∈℘(A). Then we know B⊆A.

Proof

Since X∈℘(Y), we know X⊆Y. Using this and a∈X, we conclude a∈Y.

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