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Theorem isfin5-2 8033
Description: Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
isfin5-2  |-  ( A  e.  V  ->  ( A  e. FinV 
<->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A ) ) ) )

Proof of Theorem isfin5-2
StepHypRef Expression
1 nne 2463 . . . . 5  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
21bicomi 193 . . . 4  |-  ( A  =  (/)  <->  -.  A  =/=  (/) )
32a1i 10 . . 3  |-  ( A  e.  V  ->  ( A  =  (/)  <->  -.  A  =/=  (/) ) )
4 cdadom3 7830 . . . . 5  |-  ( ( A  e.  V  /\  A  e.  V )  ->  A  ~<_  ( A  +c  A ) )
54anidms 626 . . . 4  |-  ( A  e.  V  ->  A  ~<_  ( A  +c  A
) )
6 brsdom 6900 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  <->  ( A  ~<_  ( A  +c  A )  /\  -.  A  ~~  ( A  +c  A
) ) )
76baib 871 . . . 4  |-  ( A  ~<_  ( A  +c  A
)  ->  ( A  ~<  ( A  +c  A
)  <->  -.  A  ~~  ( A  +c  A
) ) )
85, 7syl 15 . . 3  |-  ( A  e.  V  ->  ( A  ~<  ( A  +c  A )  <->  -.  A  ~~  ( A  +c  A
) ) )
93, 8orbi12d 690 . 2  |-  ( A  e.  V  ->  (
( A  =  (/)  \/  A  ~<  ( A  +c  A ) )  <->  ( -.  A  =/=  (/)  \/  -.  A  ~~  ( A  +c  A
) ) ) )
10 isfin5 7941 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
11 ianor 474 . 2  |-  ( -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  <->  ( -.  A  =/=  (/)  \/  -.  A  ~~  ( A  +c  A
) ) )
129, 10, 113bitr4g 279 1  |-  ( A  e.  V  ->  ( A  e. FinV 
<->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   class class class wbr 4039  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878    +c ccda 7809  FinVcfin5 7924
This theorem is referenced by:  fin45  8034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-cda 7810  df-fin5 7931
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