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Theorem exsnrex 3534
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 vex 2661 . . . . . 6  |-  x  e. 
_V
21snid 3524 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2179 . . . . 5  |-  ( M  =  { x }  ->  ( x  e.  M  <->  x  e.  { x }
) )
42, 3mpbiri 167 . . . 4  |-  ( M  =  { x }  ->  x  e.  M )
54pm4.71ri 387 . . 3  |-  ( M  =  { x }  <->  ( x  e.  M  /\  M  =  { x } ) )
65exbii 1567 . 2  |-  ( E. x  M  =  {
x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
7 df-rex 2397 . 2  |-  ( E. x  e.  M  M  =  { x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
86, 7bitr4i 186 1  |-  ( E. x  M  =  {
x }  <->  E. x  e.  M  M  =  { x } )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392   {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sn 3501
This theorem is referenced by: (None)
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