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Theorem en3i 6900
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
Hypotheses
Ref Expression
en3i.1  |-  A  e. 
_V
en3i.2  |-  B  e. 
_V
en3i.3  |-  ( x  e.  A  ->  C  e.  B )
en3i.4  |-  ( y  e.  B  ->  D  e.  A )
en3i.5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <-> 
y  =  C ) )
Assertion
Ref Expression
en3i  |-  A  ~~  B
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D
Allowed substitution hints:    C( x)    D( y)

Proof of Theorem en3i
StepHypRef Expression
1 en3i.1 . . . 4  |-  A  e. 
_V
21a1i 10 . . 3  |-  (  T. 
->  A  e.  _V )
3 en3i.2 . . . 4  |-  B  e. 
_V
43a1i 10 . . 3  |-  (  T. 
->  B  e.  _V )
5 en3i.3 . . . 4  |-  ( x  e.  A  ->  C  e.  B )
65a1i 10 . . 3  |-  (  T. 
->  ( x  e.  A  ->  C  e.  B ) )
7 en3i.4 . . . 4  |-  ( y  e.  B  ->  D  e.  A )
87a1i 10 . . 3  |-  (  T. 
->  ( y  e.  B  ->  D  e.  A ) )
9 en3i.5 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <-> 
y  =  C ) )
109a1i 10 . . 3  |-  (  T. 
->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
112, 4, 6, 8, 10en3d 6898 . 2  |-  (  T. 
->  A  ~~  B )
1211trud 1314 1  |-  A  ~~  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    ~~ cen 6860
This theorem is referenced by:  xpmapenlem  7028  nn0ennn  11041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
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