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Theorem isfin4 8161
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 3422 . . . . 5  |-  ( x  =  A  ->  (
y  C.  x  <->  y  C.  A ) )
2 breq2 4203 . . . . 5  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
31, 2anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( y  C.  x  /\  y  ~~  x )  <-> 
( y  C.  A  /\  y  ~~  A ) ) )
43exbidv 1636 . . 3  |-  ( x  =  A  ->  ( E. y ( y  C.  x  /\  y  ~~  x
)  <->  E. y ( y 
C.  A  /\  y  ~~  A ) ) )
54notbid 286 . 2  |-  ( x  =  A  ->  ( -.  E. y ( y 
C.  x  /\  y  ~~  x )  <->  -.  E. y
( y  C.  A  /\  y  ~~  A ) ) )
6 df-fin4 8151 . 2  |- FinIV  =  {
x  |  -.  E. y ( y  C.  x  /\  y  ~~  x
) }
75, 6elab2g 3071 1  |-  ( A  e.  V  ->  ( A  e. FinIV 
<->  -.  E. y ( y  C.  A  /\  y  ~~  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    C. wpss 3308   class class class wbr 4199    ~~ cen 7092  FinIVcfin4 8144
This theorem is referenced by:  fin4i  8162  fin4en1  8173  ssfin4  8174  infpssALT  8177  isfin4-2  8178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-fin4 8151
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