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Theorem en2i 6632
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
Hypotheses
Ref Expression
en2i.1  |-  A  e. 
_V
en2i.2  |-  B  e. 
_V
en2i.3  |-  ( x  e.  A  ->  C  e.  _V )
en2i.4  |-  ( y  e.  B  ->  D  e.  _V )
en2i.5  |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
)
Assertion
Ref Expression
en2i  |-  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 en2i
StepHypRef Expression
1 en2i.1 . . . 4  |-  A  e. 
_V
21a1i 9 . . 3  |-  ( T. 
->  A  e.  _V )
3 en2i.2 . . . 4  |-  B  e. 
_V
43a1i 9 . . 3  |-  ( T. 
->  B  e.  _V )
5 en2i.3 . . . 4  |-  ( x  e.  A  ->  C  e.  _V )
65a1i 9 . . 3  |-  ( T. 
->  ( x  e.  A  ->  C  e.  _V )
)
7 en2i.4 . . . 4  |-  ( y  e.  B  ->  D  e.  _V )
87a1i 9 . . 3  |-  ( T. 
->  ( y  e.  B  ->  D  e.  _V )
)
9 en2i.5 . . . 4  |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
)
109a1i 9 . . 3  |-  ( T. 
->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
112, 4, 6, 8, 10en2d 6630 . 2  |-  ( T. 
->  A  ~~  B )
1211mptru 1325 1  |-  A  ~~  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   T. wtru 1317    e. wcel 1465   _Vcvv 2660   class class class wbr 3899    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-en 6603
This theorem is referenced by:  mapsnen  6673  xpsnen  6683  xpassen  6692
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