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Theorem fin4i 7924
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 7923 . . 3  |-  ( A  e. FinIV  ->  ( A  e. FinIV  <->  -.  E. x ( x  C.  A  /\  x  ~~  A
) ) )
21ibi 232 . 2  |-  ( A  e. FinIV  ->  -.  E. x
( x  C.  A  /\  x  ~~  A ) )
3 relen 6868 . . . . 5  |-  Rel  ~~
43brrelexi 4729 . . . 4  |-  ( X 
~~  A  ->  X  e.  _V )
54adantl 452 . . 3  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  X  e.  _V )
6 psseq1 3263 . . . . 5  |-  ( x  =  X  ->  (
x  C.  A  <->  X  C.  A ) )
7 breq1 4026 . . . . 5  |-  ( x  =  X  ->  (
x  ~~  A  <->  X  ~~  A ) )
86, 7anbi12d 691 . . . 4  |-  ( x  =  X  ->  (
( x  C.  A  /\  x  ~~  A )  <-> 
( X  C.  A  /\  X  ~~  A ) ) )
98spcegv 2869 . . 3  |-  ( X  e.  _V  ->  (
( X  C.  A  /\  X  ~~  A )  ->  E. x ( x 
C.  A  /\  x  ~~  A ) ) )
105, 9mpcom 32 . 2  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  E. x ( x  C.  A  /\  x  ~~  A
) )
112, 10nsyl3 111 1  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  -.  A  e. FinIV )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C. wpss 3153   class class class wbr 4023    ~~ cen 6860  FinIVcfin4 7906
This theorem is referenced by:  fin4en1  7935  ssfin4  7936  ominf4  7938  isfin4-3  7941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-en 6864  df-fin4 7913
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