Theorem

Let x and A be objects. Then we have A⊆{x}∪A.

Background

The following background is necessary to understand this theorem.

(.) Let x and y be objects. Then {x}∪y is an object.

(.) 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 x, A and y be objects. Assume y∈A. Then we know y∈{x}∪A.

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

Proof

Clearly, for all a ∈ A a∈{x}∪A. Using this, we conclude A⊆{x}∪A.

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