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Theorem exsnrex 4040
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
StepHypRef Expression
1 ssnid 4031 . . . . 5  |-  x  e. 
{ x }
2 eleq2 2502 . . . . 5  |-  ( M  =  { x }  ->  ( x  e.  M  <->  x  e.  { x }
) )
31, 2mpbiri 236 . . . 4  |-  ( M  =  { x }  ->  x  e.  M )
43pm4.71ri 637 . . 3  |-  ( M  =  { x }  <->  ( x  e.  M  /\  M  =  { x } ) )
54exbii 1714 . 2  |-  ( E. x  M  =  {
x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
6 df-rex 2788 . 2  |-  ( E. x  e.  M  M  =  { x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
75, 6bitr4i 255 1  |-  ( E. x  M  =  {
x }  <->  E. x  e.  M  M  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   E.wrex 2783   {csn 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rex 2788  df-v 3089  df-sn 4003
This theorem is referenced by:  frgrawopreg1  25623  frgrawopreg2  25624
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