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Theorem exsnrex 3441
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 2577 . . . . . 6  |-  x  e. 
_V
21snid 3430 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2117 . . . . 5  |-  ( M  =  { x }  ->  ( x  e.  M  <->  x  e.  { x }
) )
42, 3mpbiri 161 . . . 4  |-  ( M  =  { x }  ->  x  e.  M )
54pm4.71ri 378 . . 3  |-  ( M  =  { x }  <->  ( x  e.  M  /\  M  =  { x } ) )
65exbii 1512 . 2  |-  ( E. x  M  =  {
x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
7 df-rex 2329 . 2  |-  ( E. x  e.  M  M  =  { x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
86, 7bitr4i 180 1  |-  ( E. x  M  =  {
x }  <->  E. x  e.  M  M  =  { x } )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sn 3409
This theorem is referenced by: (None)
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