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Theorem fin4i 8591
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 8590 . . 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 7440 . . . . 5  |-  Rel  ~~
43brrelexi 4954 . . . 4  |-  ( X 
~~  A  ->  X  e.  _V )
54adantl 464 . . 3  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  X  e.  _V )
6 psseq1 3505 . . . . 5  |-  ( x  =  X  ->  (
x  C.  A  <->  X  C.  A
) )
7 breq1 4370 . . . . 5  |-  ( x  =  X  ->  (
x  ~~  A  <->  X  ~~  A ) )
86, 7anbi12d 708 . . . 4  |-  ( x  =  X  ->  (
( x  C.  A  /\  x  ~~  A )  <-> 
( X  C.  A  /\  X  ~~  A ) ) )
98spcegv 3120 . . 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 367    = wceq 1399   E.wex 1620    e. wcel 1826   _Vcvv 3034    C. wpss 3390   class class class wbr 4367    ~~ cen 7432  FinIVcfin4 8573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-en 7436  df-fin4 8580
This theorem is referenced by:  fin4en1  8602  ssfin4  8603  ominf4  8605  isfin4-3  8608
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