Theorem

Let A and B be objects. Then we have A∩B⊆A.

Background

The following background is necessary to understand this theorem.

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

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

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

Proof

We know for all x x∈A∩B implies x∈A. Using this, we conclude A∩B⊆A.

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