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Theorem rext 4561
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 ssnid 3927 . . 3  |-  x  e. 
{ x }
2 snex 4554 . . . 4  |-  { x }  e.  _V
3 eleq2 2504 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
4 eleq2 2504 . . . . 5  |-  ( z  =  { x }  ->  ( y  e.  z  <-> 
y  e.  { x } ) )
53, 4imbi12d 320 . . . 4  |-  ( z  =  { x }  ->  ( ( x  e.  z  ->  y  e.  z )  <->  ( x  e.  { x }  ->  y  e.  { x }
) ) )
62, 5spcv 3084 . . 3  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  (
x  e.  { x }  ->  y  e.  {
x } ) )
71, 6mpi 17 . 2  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  y  e.  { x } )
8 elsn 3912 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
9 equcomi 1731 . . 3  |-  ( y  =  x  ->  x  =  y )
108, 9sylbi 195 . 2  |-  ( y  e.  { x }  ->  x  =  y )
117, 10syl 16 1  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    = wceq 1369    e. wcel 1756   {csn 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-v 2995  df-dif 3352  df-un 3354  df-nul 3659  df-sn 3899  df-pr 3901
This theorem is referenced by: (None)
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