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Theorem en3i 6765
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 9 . . 3  |-  ( T. 
->  A  e.  _V )
3 en3i.2 . . . 4  |-  B  e. 
_V
43a1i 9 . . 3  |-  ( T. 
->  B  e.  _V )
5 en3i.3 . . . 4  |-  ( x  e.  A  ->  C  e.  B )
65a1i 9 . . 3  |-  ( T. 
->  ( x  e.  A  ->  C  e.  B ) )
7 en3i.4 . . . 4  |-  ( y  e.  B  ->  D  e.  A )
87a1i 9 . . 3  |-  ( T. 
->  ( y  e.  B  ->  D  e.  A ) )
9 en3i.5 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <-> 
y  =  C ) )
109a1i 9 . . 3  |-  ( T. 
->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
112, 4, 6, 8, 10en3d 6763 . 2  |-  ( T. 
->  A  ~~  B )
1211mptru 1362 1  |-  A  ~~  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   T. wtru 1354    e. wcel 2148   _Vcvv 2737   class class class wbr 4000    ~~ cen 6732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-en 6735
This theorem is referenced by:  xpmapenlem  6843  nn0ennn  10419  oddennn  12376  evenennn  12377  znnen  12382
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