MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin4i Structured version   Unicode version

Theorem fin4i 8455
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 8454 . . 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 7303 . . . . 5  |-  Rel  ~~
43brrelexi 4866 . . . 4  |-  ( X 
~~  A  ->  X  e.  _V )
54adantl 463 . . 3  |-  ( ( X  C.  A  /\  X  ~~  A )  ->  X  e.  _V )
6 psseq1 3431 . . . . 5  |-  ( x  =  X  ->  (
x  C.  A  <->  X  C.  A
) )
7 breq1 4283 . . . . 5  |-  ( x  =  X  ->  (
x  ~~  A  <->  X  ~~  A ) )
86, 7anbi12d 703 . . . 4  |-  ( x  =  X  ->  (
( x  C.  A  /\  x  ~~  A )  <-> 
( X  C.  A  /\  X  ~~  A ) ) )
98spcegv 3047 . . 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 1362   E.wex 1589    e. wcel 1755   _Vcvv 2962    C. wpss 3317   class class class wbr 4280    ~~ cen 7295  FinIVcfin4 8437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-xp 4833  df-rel 4834  df-en 7299  df-fin4 8444
This theorem is referenced by:  fin4en1  8466  ssfin4  8467  ominf4  8469  isfin4-3  8472
  Copyright terms: Public domain W3C validator