Theorem exsnrex 4065

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.)

Assertion

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

Proof of Theorem exsnrex

Step	Hyp	Ref	Expression
1		ssnid 4056	. . . . 5 |- x e. { x }
2		eleq2 2540	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
3	1, 2	mpbiri 233	. . . 4 |- ( M = { x } -> x e. M )
4	3	pm4.71ri 633	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
5	4	exbii 1644	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
6		df-rex 2820	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7	5, 6	bitr4i 252	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   E.wrex 2815   {csn 4027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-v 3115  df-sn 4028

This theorem is referenced by:  frgrawopreg1  24755  frgrawopreg2  24756