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Theorem idssen 7621
Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
idssen  |-  _I  C_  ~~

Proof of Theorem idssen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 4982 . 2  |-  Rel  _I
2 vex 3090 . . . . 5  |-  y  e. 
_V
32ideq 5007 . . . 4  |-  ( x  _I  y  <->  x  =  y )
4 vex 3090 . . . . 5  |-  x  e. 
_V
5 eqeng 7610 . . . . 5  |-  ( x  e.  _V  ->  (
x  =  y  ->  x  ~~  y ) )
64, 5ax-mp 5 . . . 4  |-  ( x  =  y  ->  x  ~~  y )
73, 6sylbi 198 . . 3  |-  ( x  _I  y  ->  x  ~~  y )
8 df-br 4427 . . 3  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
9 df-br 4427 . . 3  |-  ( x 
~~  y  <->  <. x ,  y >.  e.  ~~  )
107, 8, 93imtr3i 268 . 2  |-  ( <.
x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ~~  )
111, 10relssi 4946 1  |-  _I  C_  ~~
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1870   _Vcvv 3087    C_ wss 3442   <.cop 4008   class class class wbr 4426    _I cid 4764    ~~ cen 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-en 7578
This theorem is referenced by: (None)
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