Theorem exsnrex 3413

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 2560	. . . . . 6 |- x e. _V
2	1	snid 3402	. . . . 5 |- x e. { x }
3		eleq2 2101	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
4	2, 3	mpbiri 157	. . . 4 |- ( M = { x } -> x e. M )
5	4	pm4.71ri 372	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
6	5	exbii 1496	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7		df-rex 2312	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
8	6, 7	bitr4i 176	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sn 3381

This theorem is referenced by: (None)