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Theorem rext 4688
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 4049 . . 3  |-  x  e. 
{ x }
2 snex 4681 . . . 4  |-  { x }  e.  _V
3 eleq2 2533 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
4 eleq2 2533 . . . . 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 3197 . . 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 4034 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
9 equcomi 1737 . . 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 1372    = wceq 1374    e. wcel 1762   {csn 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-dif 3472  df-un 3474  df-nul 3779  df-sn 4021  df-pr 4023
This theorem is referenced by: (None)
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