Theorem

Let x, y, z and u be objects. Assume {{x},{x,y}}={{z},{z,u}}. Then we have x=z.

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

(.) We use {x1,...,xn} as shorthand notation for the finite set with elements x1, ..., xn.

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 and y be objects. Then we know x∈{x}∪y.

(.) Let x and y be objects. Let φ(x) be a proposition depending on an object x. Assume x=y and φ(y). Then we know φ(x).

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

Proof

It is enough to show {z}∈{{x},{x,y}}. We have {z}∈{{z},{z,u}}. Using this and {{x},{x,y}}={{z},{z,u}}, we are done.

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