Theorem

Let A be an object. Let X and Y be members of ℘(A). Then we have A \ X∈℘(A \ X∩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.

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

(.) 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. Then "A⊆B" is a proposition.

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

(.) Let A be an object. Let X and Y be members of ℘(A). Then we know A \ X⊆A \ X∩Y.

Proof

Clearly, A \ X⊆A \ X∩Y. Using this, we conclude A \ X∈℘(A \ X∩Y).

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