Theorem

Let A be an object. Let X, Y and Z be members of ℘(A). Assume Y⊆Z. Then we have X∩Y⊆Z.

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.

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

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

Background for Proof

The following background is necessary for the proof.

(.) Let A and B be objects. Assume for all a ∈ A a∈B. Then we know A⊆B.

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

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

Proof

It is enough to show for all x ∈ X∩Y x∈Z. Let x be a member of X∩Y. We know x∈Y. Using this and Y⊆Z, we conclude x∈Z.

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