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Theorem idssen 6424
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 4523 . 2  |-  Rel  _I
2 vex 2615 . . . . 5  |-  y  e. 
_V
32ideq 4546 . . . 4  |-  ( x  _I  y  <->  x  =  y )
4 vex 2615 . . . . 5  |-  x  e. 
_V
5 eqeng 6413 . . . . 5  |-  ( x  e.  _V  ->  (
x  =  y  ->  x  ~~  y ) )
64, 5ax-mp 7 . . . 4  |-  ( x  =  y  ->  x  ~~  y )
73, 6sylbi 119 . . 3  |-  ( x  _I  y  ->  x  ~~  y )
8 df-br 3812 . . 3  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
9 df-br 3812 . . 3  |-  ( x 
~~  y  <->  <. x ,  y >.  e.  ~~  )
107, 8, 93imtr3i 198 . 2  |-  ( <.
x ,  y >.  e.  _I  ->  <. x ,  y >.  e.  ~~  )
111, 10relssi 4487 1  |-  _I  C_  ~~
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   _Vcvv 2612    C_ wss 2984   <.cop 3425   class class class wbr 3811    _I cid 4079    ~~ cen 6385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-en 6388
This theorem is referenced by: (None)
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