Theorem eusnsn

Description: There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022)

	Ref	Expression
Assertion	eusnsn	|- E! x { x } = { y }

Proof

Step	Hyp	Ref	Expression
1		equequ2	|- ( z = y -> ( x = z <-> x = y ) )
2	1	bibi2d	|- ( z = y -> ( ( { x } = { y } <-> x = z ) <-> ( { x } = { y } <-> x = y ) ) )
3	2	albidv	|- ( z = y -> ( A. x ( { x } = { y } <-> x = z ) <-> A. x ( { x } = { y } <-> x = y ) ) )
4		sneqbg	|- ( x e. _V -> ( { x } = { y } <-> x = y ) )
5	4	elv	|- ( { x } = { y } <-> x = y )
6	5	ax-gen	|- A. x ( { x } = { y } <-> x = y )
7	3 6	speivw	|- E. z A. x ( { x } = { y } <-> x = z )
8		eu6	|- ( E! x { x } = { y } <-> E. z A. x ( { x } = { y } <-> x = z ) )
9	7 8	mpbir	|- E! x { x } = { y }