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Theorem isfin4 8466
Description: Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin4  |-  ( A  e.  V  ->  ( A  e. FinIV 
<->  -.  E. y ( y  C.  A  /\  y  ~~  A ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 psseq2 3444 . . . . 5  |-  ( x  =  A  ->  (
y  C.  x  <->  y  C.  A
) )
2 breq2 4296 . . . . 5  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
31, 2anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( y  C.  x  /\  y  ~~  x )  <-> 
( y  C.  A  /\  y  ~~  A ) ) )
43exbidv 1680 . . 3  |-  ( x  =  A  ->  ( E. y ( y  C.  x  /\  y  ~~  x
)  <->  E. y ( y 
C.  A  /\  y  ~~  A ) ) )
54notbid 294 . 2  |-  ( x  =  A  ->  ( -.  E. y ( y 
C.  x  /\  y  ~~  x )  <->  -.  E. y
( y  C.  A  /\  y  ~~  A ) ) )
6 df-fin4 8456 . 2  |- FinIV  =  {
x  |  -.  E. y ( y  C.  x  /\  y  ~~  x
) }
75, 6elab2g 3108 1  |-  ( A  e.  V  ->  ( A  e. FinIV 
<->  -.  E. y ( y  C.  A  /\  y  ~~  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    C. wpss 3329   class class class wbr 4292    ~~ cen 7307  FinIVcfin4 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-fin4 8456
This theorem is referenced by:  fin4i  8467  fin4en1  8478  ssfin4  8479  infpssALT  8482  isfin4-2  8483
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