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Theorem isfin7-2 8038
Description: A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
isfin7-2  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  ( A  e.  dom  card 
->  A  e.  Fin ) ) )

Proof of Theorem isfin7-2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfin7 7943 . . . 4  |-  ( A  e. FinVII  ->  ( A  e. FinVII  <->  -.  E. x  e.  ( On 
\  om ) A 
~~  x ) )
21ibi 232 . . 3  |-  ( A  e. FinVII  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
3 isnum2 7594 . . . . 5  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
4 ensym 6926 . . . . . . . . 9  |-  ( x 
~~  A  ->  A  ~~  x )
5 simprl 732 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  On )
6 enfi 7095 . . . . . . . . . . . . . . 15  |-  ( A 
~~  x  ->  ( A  e.  Fin  <->  x  e.  Fin ) )
7 onfin 7067 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  (
x  e.  Fin  <->  x  e.  om ) )
86, 7sylan9bbr 681 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( A  e.  Fin  <->  x  e.  om ) )
98biimprd 214 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( x  e.  om  ->  A  e.  Fin )
)
109con3d 125 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( -.  A  e. 
Fin  ->  -.  x  e.  om ) )
1110impcom 419 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  -.  x  e.  om )
12 eldif 3175 . . . . . . . . . . 11  |-  ( x  e.  ( On  \  om )  <->  ( x  e.  On  /\  -.  x  e.  om ) )
135, 11, 12sylanbrc 645 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  ( On  \  om )
)
14 simprr 733 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  A  ~~  x )
1513, 14jca 518 . . . . . . . . 9  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
164, 15sylanr2 634 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  x  ~~  A ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
1716ex 423 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  ( ( x  e.  On  /\  x  ~~  A )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) ) )
1817reximdv2 2665 . . . . . 6  |-  ( -.  A  e.  Fin  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
1918com12 27 . . . . 5  |-  ( E. x  e.  On  x  ~~  A  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On 
\  om ) A 
~~  x ) )
203, 19sylbi 187 . . . 4  |-  ( A  e.  dom  card  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
2120con1d 116 . . 3  |-  ( A  e.  dom  card  ->  ( -.  E. x  e.  ( On  \  om ) A  ~~  x  ->  A  e.  Fin )
)
222, 21syl5com 26 . 2  |-  ( A  e. FinVII  ->  ( A  e. 
dom  card  ->  A  e.  Fin ) )
23 eldifi 3311 . . . . . . 7  |-  ( x  e.  ( On  \  om )  ->  x  e.  On )
24 ensym 6926 . . . . . . 7  |-  ( A 
~~  x  ->  x  ~~  A )
25 isnumi 7595 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
2623, 24, 25syl2an 463 . . . . . 6  |-  ( ( x  e.  ( On 
\  om )  /\  A  ~~  x )  ->  A  e.  dom  card )
2726rexlimiva 2675 . . . . 5  |-  ( E. x  e.  ( On 
\  om ) A 
~~  x  ->  A  e.  dom  card )
2827con3i 127 . . . 4  |-  ( -.  A  e.  dom  card  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
29 isfin7 7943 . . . 4  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. x  e.  ( On  \  om ) A  ~~  x ) )
3028, 29syl5ibr 212 . . 3  |-  ( A  e.  V  ->  ( -.  A  e.  dom  card 
->  A  e. FinVII ) )
31 fin17 8036 . . . 4  |-  ( A  e.  Fin  ->  A  e. FinVII )
3231a1i 10 . . 3  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  A  e. FinVII ) )
3330, 32jad 154 . 2  |-  ( A  e.  V  ->  (
( A  e.  dom  card 
->  A  e.  Fin )  ->  A  e. FinVII ) )
3422, 33impbid2 195 1  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  ( A  e.  dom  card 
->  A  e.  Fin ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   E.wrex 2557    \ cdif 3162   class class class wbr 4039   Oncon0 4408   omcom 4672   dom cdm 4705    ~~ cen 6876   Fincfn 6879   cardccrd 7584  FinVIIcfin7 7926
This theorem is referenced by:  fin71num  8039  dffin7-2  8040
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-lim 4413  df-suc 4414  df-om 4673  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-fin7 7933
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