Theorem exsnrex 3625

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		vex 2733	. . . . . 6 |- x e. _V
2	1	snid 3614	. . . . 5 |- x e. { x }
3		eleq2 2234	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
4	2, 3	mpbiri 167	. . . 4 |- ( M = { x } -> x e. M )
5	4	pm4.71ri 390	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
6	5	exbii 1598	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7		df-rex 2454	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
8	6, 7	bitr4i 186	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sn 3589

This theorem is referenced by: (None)