Theorem exsnrex 3711

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 2805	. . . . . 6 |- x e. _V
2	1	snid 3700	. . . . 5 |- x e. { x }
3		eleq2 2295	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
4	2, 3	mpbiri 168	. . . 4 |- ( M = { x } -> x e. M )
5	4	pm4.71ri 392	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
6	5	exbii 1653	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7		df-rex 2516	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
8	6, 7	bitr4i 187	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   E.wrex 2511   {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sn 3675

This theorem is referenced by: (None)