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Theorem idssen 7152
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 5002 . 2  |-  Rel  _I
2 vex 2959 . . . . 5  |-  y  e. 
_V
32ideq 5025 . . . 4  |-  ( x  _I  y  <->  x  =  y )
4 vex 2959 . . . . 5  |-  x  e. 
_V
5 eqeng 7141 . . . . 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 4213 . . 3  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
9 df-br 4213 . . 3  |-  ( x 
~~  y  <->  <. x ,  y >.  e.  ~~  )
107, 8, 93imtr3i 257 . 2  |-  ( <.
x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ~~  )
111, 10relssi 4967 1  |-  _I  C_  ~~
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2956    C_ wss 3320   <.cop 3817   class class class wbr 4212    _I cid 4493    ~~ cen 7106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-en 7110
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