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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 2791 . . . 4  |-  x  e. 
_V
21snid 3667 . . 3  |-  x  e. 
{ x }
3 snex 4216 . . . 4  |-  { x }  e.  _V
4 eleq2 2344 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
5 eleq2 2344 . . . . 5  |-  ( z  =  { x }  ->  ( y  e.  z  <-> 
y  e.  { x } ) )
64, 5imbi12d 311 . . . 4  |-  ( z  =  { x }  ->  ( ( x  e.  z  ->  y  e.  z )  <->  ( x  e.  { x }  ->  y  e.  { x }
) ) )
73, 6spcv 2874 . . 3  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  (
x  e.  { x }  ->  y  e.  {
x } ) )
82, 7mpi 16 . 2  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  y  e.  { x } )
9 elsn 3655 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
10 equcomi 1646 . . 3  |-  ( y  =  x  ->  x  =  y )
119, 10sylbi 187 . 2  |-  ( y  e.  { x }  ->  x  =  y )
128, 11syl 15 1  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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