Theorem exsnrex 4040

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 4031	. . . . 5 |- x e. { x }
2		eleq2 2502	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
3	1, 2	mpbiri 236	. . . 4 |- ( M = { x } -> x e. M )
4	3	pm4.71ri 637	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
5	4	exbii 1714	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
6		df-rex 2788	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7	5, 6	bitr4i 255	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   e. wcel 1870   E.wrex 2783   {csn 4002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rex 2788  df-v 3089  df-sn 4003

This theorem is referenced by:  frgrawopreg1  25623  frgrawopreg2  25624