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Theorem en3i 6339
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 6337 . 2  |-  ( T. 
->  A  ~~  B )
1211trud 1294 1  |-  A  ~~  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   T. wtru 1286    e. wcel 1434   _Vcvv 2610   class class class wbr 3805    ~~ cen 6306
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 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
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 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-en 6309
This theorem is referenced by:  nn0ennn  9567  oddennn  10812  evenennn  10813  znnen  10818
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