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Theorem rext 4222
Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
Assertion
Ref Expression
rext  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Distinct variable group:    x, y, z

Proof of Theorem rext
StepHypRef Expression
1 vex 2793 . . . 4  |-  x  e. 
_V
21snid 3669 . . 3  |-  x  e. 
{ x }
3 snex 4216 . . . 4  |-  { x }  e.  _V
4 eleq2 2346 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
5 eleq2 2346 . . . . 5  |-  ( z  =  { x }  ->  ( y  e.  z  <-> 
y  e.  { x } ) )
64, 5imbi12d 313 . . . 4  |-  ( z  =  { x }  ->  ( ( x  e.  z  ->  y  e.  z )  <->  ( x  e.  { x }  ->  y  e.  { x }
) ) )
73, 6spcv 2876 . . 3  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  (
x  e.  { x }  ->  y  e.  {
x } ) )
82, 7mpi 18 . 2  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  y  e.  { x } )
9 elsn 3657 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
10 equcomi 1647 . . 3  |-  ( y  =  x  ->  x  =  y )
119, 10sylbi 189 . 2  |-  ( y  e.  { x }  ->  x  =  y )
128, 11syl 17 1  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1528    = wceq 1624    e. wcel 1685   {csn 3642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-un 3159  df-nul 3458  df-sn 3648  df-pr 3649
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