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Theorem fin4i 8568
Description: Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
fin4i  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  -.  A  e. FinIV )

Proof of Theorem fin4i
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfin4 8567 . . 3  |-  ( A  e. FinIV  ->  ( A  e. FinIV  <->  -.  E. x ( x  C.  A  /\  x  ~~  A
) ) )
21ibi 241 . 2  |-  ( A  e. FinIV  ->  -.  E. x
( x  C.  A  /\  x  ~~  A ) )
3 relen 7415 . . . . 5  |-  Rel  ~~
43brrelexi 4977 . . . 4  |-  ( X 
~~  A  ->  X  e.  _V )
54adantl 466 . . 3  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  X  e.  _V )
6 psseq1 3541 . . . . 5  |-  ( x  =  X  ->  (
x  C.  A  <->  X  C.  A
) )
7 breq1 4393 . . . . 5  |-  ( x  =  X  ->  (
x  ~~  A  <->  X  ~~  A ) )
86, 7anbi12d 710 . . . 4  |-  ( x  =  X  ->  (
( x  C.  A  /\  x  ~~  A )  <-> 
( X  C.  A  /\  X  ~~  A ) ) )
98spcegv 3154 . . 3  |-  ( X  e.  _V  ->  (
( X  C.  A  /\  X  ~~  A )  ->  E. x ( x 
C.  A  /\  x  ~~  A ) ) )
105, 9mpcom 36 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. x ( x  C.  A  /\  x  ~~  A
) )
112, 10nsyl3 119 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  -.  A  e. FinIV )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3068    C. wpss 3427   class class class wbr 4390    ~~ cen 7407  FinIVcfin4 8550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-xp 4944  df-rel 4945  df-en 7411  df-fin4 8557
This theorem is referenced by:  fin4en1  8579  ssfin4  8580  ominf4  8582  isfin4-3  8585
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