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Theorem rext 4404
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 2951 . . . 4  |-  x  e. 
_V
21snid 3833 . . 3  |-  x  e. 
{ x }
3 snex 4397 . . . 4  |-  { x }  e.  _V
4 eleq2 2496 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
5 eleq2 2496 . . . . 5  |-  ( z  =  { x }  ->  ( y  e.  z  <-> 
y  e.  { x } ) )
64, 5imbi12d 312 . . . 4  |-  ( z  =  { x }  ->  ( ( x  e.  z  ->  y  e.  z )  <->  ( x  e.  { x }  ->  y  e.  { x }
) ) )
73, 6spcv 3034 . . 3  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  (
x  e.  { x }  ->  y  e.  {
x } ) )
82, 7mpi 17 . 2  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  y  e.  { x } )
9 elsn 3821 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
10 equcomi 1691 . . 3  |-  ( y  =  x  ->  x  =  y )
119, 10sylbi 188 . 2  |-  ( y  e.  { x }  ->  x  =  y )
128, 11syl 16 1  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   {csn 3806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-pr 3813
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