Theorem exsnrex 3664

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 2766	. . . . . 6 |- x e. _V
2	1	snid 3653	. . . . 5 |- x e. { x }
3		eleq2 2260	. . . . 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 1619	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7		df-rex 2481	. 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 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sn 3628

This theorem is referenced by: (None)