Theorem

Let A, B and x be objects. Assume x∈B. Then we have x∈A∪B.

Background

The following background is necessary to understand this theorem.

(.) Let A and x be objects. Then x∈A 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.

(.) ∅ is an object.

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

(.) Let A be an object. Then UA is an object.

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

(.) Let A and B be objects. Then A∪B is the object given by U({A,B}).

(.) Let x and y be objects. Then we know y∈{x,y}.

Proof

It is enough to show x∈U({A,B}). We know B∈{A,B}. Using this and x∈B, we are done.

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