Theorem exsnrex 4000

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 3989	. . . . 5 |- x e. { x }
2		eleq2 2538	. . . . 5 |- ( M = { x } -> ( x e. M <-> x e. { x } ) )
3	1, 2	mpbiri 241	. . . 4 |- ( M = { x } -> x e. M )
4	3	pm4.71ri 645	. . 3 |- ( M = { x } <-> ( x e. M /\ M = { x } ) )
5	4	exbii 1726	. 2 |- ( E. x M = { x } <-> E. x ( x e. M /\ M = { x } ) )
6		df-rex 2762	. 2 |- ( E. x e. M M = { x } <-> E. x ( x e. M /\ M = { x } ) )
7	5, 6	bitr4i 260	1 |- ( E. x M = { x } <-> E. x e. M M = { x } )

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   E.wrex 2757   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sn 3960

This theorem is referenced by:  frgrawopreg1  25857  frgrawopreg2  25858