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Theorem en3i 7547
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 11 . . 3  |-  ( T. 
->  A  e.  _V )
3 en3i.2 . . . 4  |-  B  e. 
_V
43a1i 11 . . 3  |-  ( T. 
->  B  e.  _V )
5 en3i.3 . . . 4  |-  ( x  e.  A  ->  C  e.  B )
65a1i 11 . . 3  |-  ( T. 
->  ( x  e.  A  ->  C  e.  B ) )
7 en3i.4 . . . 4  |-  ( y  e.  B  ->  D  e.  A )
87a1i 11 . . 3  |-  ( T. 
->  ( y  e.  B  ->  D  e.  A ) )
9 en3i.5 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <-> 
y  =  C ) )
109a1i 11 . . 3  |-  ( T. 
->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
112, 4, 6, 8, 10en3d 7545 . 2  |-  ( T. 
->  A  ~~  B )
1211trud 1407 1  |-  A  ~~  B
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   T. wtru 1399    e. wcel 1823   _Vcvv 3106   class class class wbr 4439    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-en 7510
This theorem is referenced by:  xpmapenlem  7677  nn0ennn  12074
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