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Theorem idssen 7119
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 4969 . 2  |-  Rel  _I
2 vex 2927 . . . . 5  |-  y  e. 
_V
32ideq 4992 . . . 4  |-  ( x  _I  y  <->  x  =  y )
4 vex 2927 . . . . 5  |-  x  e. 
_V
5 eqeng 7108 . . . . 5  |-  ( x  e.  _V  ->  (
x  =  y  ->  x  ~~  y ) )
64, 5ax-mp 8 . . . 4  |-  ( x  =  y  ->  x  ~~  y )
73, 6sylbi 188 . . 3  |-  ( x  _I  y  ->  x  ~~  y )
8 df-br 4181 . . 3  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
9 df-br 4181 . . 3  |-  ( x 
~~  y  <->  <. x ,  y >.  e.  ~~  )
107, 8, 93imtr3i 257 . 2  |-  ( <.
x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ~~  )
111, 10relssi 4934 1  |-  _I  C_  ~~
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   _Vcvv 2924    C_ wss 3288   <.cop 3785   class class class wbr 4180    _I cid 4461    ~~ cen 7073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-en 7077
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