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Theorem fin4i 7940
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 7939 . . 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 6884 . . . . 5  |-  Rel  ~~
43brrelexi 4745 . . . 4  |-  ( X 
~~  A  ->  X  e.  _V )
54adantl 452 . . 3  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  X  e.  _V )
6 psseq1 3276 . . . . 5  |-  ( x  =  X  ->  (
x  C.  A  <->  X  C.  A ) )
7 breq1 4042 . . . . 5  |-  ( x  =  X  ->  (
x  ~~  A  <->  X  ~~  A ) )
86, 7anbi12d 691 . . . 4  |-  ( x  =  X  ->  (
( x  C.  A  /\  x  ~~  A )  <-> 
( X  C.  A  /\  X  ~~  A ) ) )
98spcegv 2882 . . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C. wpss 3166   class class class wbr 4039    ~~ cen 6876  FinIVcfin4 7922
This theorem is referenced by:  fin4en1  7951  ssfin4  7952  ominf4  7954  isfin4-3  7957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-en 6880  df-fin4 7929
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