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Theorem idssen 7570
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 5135 . 2  |-  Rel  _I
2 vex 3121 . . . . 5  |-  y  e. 
_V
32ideq 5160 . . . 4  |-  ( x  _I  y  <->  x  =  y )
4 vex 3121 . . . . 5  |-  x  e. 
_V
5 eqeng 7559 . . . . 5  |-  ( x  e.  _V  ->  (
x  =  y  ->  x  ~~  y ) )
64, 5ax-mp 5 . . . 4  |-  ( x  =  y  ->  x  ~~  y )
73, 6sylbi 195 . . 3  |-  ( x  _I  y  ->  x  ~~  y )
8 df-br 4453 . . 3  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
9 df-br 4453 . . 3  |-  ( x 
~~  y  <->  <. x ,  y >.  e.  ~~  )
107, 8, 93imtr3i 265 . 2  |-  ( <.
x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ~~  )
111, 10relssi 5099 1  |-  _I  C_  ~~
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3118    C_ wss 3481   <.cop 4038   class class class wbr 4452    _I cid 4795    ~~ cen 7523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-en 7527
This theorem is referenced by: (None)
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