Theorem exsnrex

Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018)

	Ref	Expression
Assertion	exsnrex	|- ( E. x M = { x } <-> E. x e. M M = { x } )

Proof

Step	Hyp	Ref	Expression
1		vsnid	|- x e. { x }
2		eleq2	|- ( M = { x } -> ( x e. M <-> x e. { x } ) )
3	1 2	mpbiri	|- ( M = { x } -> x e. M )
4	3	pm4.71ri	|- ( M = { x } <-> ( x e. M /\ M = { x } ) )
5	4	exbii	|- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
6		df-rex	|- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7	5 6	bitr4i	|- ( E. x M = { x } <-> E. x e. M M = { x } )