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Theorem isfin4 8680
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 3577 . . . . 5  |-  ( x  =  A  ->  (
y  C.  x  <->  y  C.  A
) )
2 breq2 4441 . . . . 5  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
31, 2anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( y  C.  x  /\  y  ~~  x )  <-> 
( y  C.  A  /\  y  ~~  A ) ) )
43exbidv 1701 . . 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 8670 . 2  |- FinIV  =  {
x  |  -.  E. y ( y  C.  x  /\  y  ~~  x
) }
75, 6elab2g 3234 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 1383   E.wex 1599    e. wcel 1804    C. wpss 3462   class class class wbr 4437    ~~ cen 7515  FinIVcfin4 8663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-fin4 8670
This theorem is referenced by:  fin4i  8681  fin4en1  8692  ssfin4  8693  infpssALT  8696  isfin4-2  8697
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