Theorem

Let A and B be objects. Assume A⊆B. 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.

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

(.) Let φ be a proposition. Then ¬φ is a proposition.

(.) Let φ and ψ be propositions. Assume φ and ¬φ. Then we know ψ.

(.) 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.

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

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

(.) Let A be an object. Assume A⊆∅. Then we know A=∅.

Proof

We will show A \ B⊆∅. It is enough to show for all x x∈A \ B implies x∈∅. Let x be an object. Assume x∈A \ B.

Claim: x∈B.

Proof of Claim: Since x∈A \ B, we know x∈A. Using this and A⊆B, we are done. This completes the proof of the claim.

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