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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
StepHypRef Expression
1 ssnid 3989 . . . . 5  |-  x  e. 
{ x }
2 eleq2 2538 . . . . 5  |-  ( M  =  { x }  ->  ( x  e.  M  <->  x  e.  { x }
) )
31, 2mpbiri 241 . . . 4  |-  ( M  =  { x }  ->  x  e.  M )
43pm4.71ri 645 . . 3  |-  ( M  =  { x }  <->  ( x  e.  M  /\  M  =  { x } ) )
54exbii 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 } ) )
75, 6bitr4i 260 1  |-  ( E. x  M  =  {
x }  <->  E. x  e.  M  M  =  { x } )
Colors of variables: wff setvar class
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
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