Theorem

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

Background

The following background is necessary to understand this theorem.

(.) 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. Then we know A⊆B.

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

Proof

It is enough to show for all x x∈A implies x∈C. Let x be an object. Assume x∈A. From this and A⊆B, we know x∈B. Using this and B⊆C, we conclude x∈C.

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