MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin4 Unicode version

Theorem isfin4 8133
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 3395 . . . . 5  |-  ( x  =  A  ->  (
y  C.  x  <->  y  C.  A ) )
2 breq2 4176 . . . . 5  |-  ( x  =  A  ->  (
y  ~~  x  <->  y  ~~  A ) )
31, 2anbi12d 692 . . . 4  |-  ( x  =  A  ->  (
( y  C.  x  /\  y  ~~  x )  <-> 
( y  C.  A  /\  y  ~~  A ) ) )
43exbidv 1633 . . 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 8123 . 2  |- FinIV  =  {
x  |  -.  E. y ( y  C.  x  /\  y  ~~  x
) }
75, 6elab2g 3044 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 1547    = wceq 1649    e. wcel 1721    C. wpss 3281   class class class wbr 4172    ~~ cen 7065  FinIVcfin4 8116
This theorem is referenced by:  fin4i  8134  fin4en1  8145  ssfin4  8146  infpssALT  8149  isfin4-2  8150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-fin4 8123
  Copyright terms: Public domain W3C validator