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Theorem en3i 6943
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 6941 . 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 1633    e. wcel 1701   _Vcvv 2822   class class class wbr 4060    ~~ cen 6903
This theorem is referenced by:  xpmapenlem  7071  nn0ennn  11088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-en 6907
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