Theorem

Let A and B be objects. Assume A⊆B and B⊆A. Then we have A=B.

Background

The following background is necessary to understand this theorem.

(.) Let x and y be objects. Then x=y is a proposition.

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

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 and for all x x∈B implies x∈A. Then we know A=B.

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

Proof

Since A⊆B, we have for all x x∈A implies x∈B. Since B⊆A, we know for all x x∈B implies x∈A. Using this and for all x x∈A implies x∈B, we conclude A=B.

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