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Theorem exsnrex 4065
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 4056 . . . . 5  |-  x  e. 
{ x }
2 eleq2 2540 . . . . 5  |-  ( M  =  { x }  ->  ( x  e.  M  <->  x  e.  { x }
) )
31, 2mpbiri 233 . . . 4  |-  ( M  =  { x }  ->  x  e.  M )
43pm4.71ri 633 . . 3  |-  ( M  =  { x }  <->  ( x  e.  M  /\  M  =  { x } ) )
54exbii 1644 . 2  |-  ( E. x  M  =  {
x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
6 df-rex 2820 . 2  |-  ( E. x  e.  M  M  =  { x }  <->  E. x
( x  e.  M  /\  M  =  {
x } ) )
75, 6bitr4i 252 1  |-  ( E. x  M  =  {
x }  <->  E. x  e.  M  M  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-v 3115  df-sn 4028
This theorem is referenced by:  frgrawopreg1  24755  frgrawopreg2  24756
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