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Theorem isfin7-2 8844
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 8749 . . . 4  |-  ( A  e. FinVII  ->  ( A  e. FinVII  <->  -.  E. x  e.  ( On 
\  om ) A 
~~  x ) )
21ibi 249 . . 3  |-  ( A  e. FinVII  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
3 isnum2 8397 . . . . 5  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
4 ensym 7636 . . . . . . . . 9  |-  ( x 
~~  A  ->  A  ~~  x )
5 simprl 772 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  On )
6 enfi 7806 . . . . . . . . . . . . . . 15  |-  ( A 
~~  x  ->  ( A  e.  Fin  <->  x  e.  Fin ) )
7 onfin 7781 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  (
x  e.  Fin  <->  x  e.  om ) )
86, 7sylan9bbr 715 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( A  e.  Fin  <->  x  e.  om ) )
98biimprd 231 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( x  e.  om  ->  A  e.  Fin )
)
109con3d 140 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( -.  A  e. 
Fin  ->  -.  x  e.  om ) )
1110impcom 437 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  -.  x  e.  om )
125, 11eldifd 3401 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  ( On  \  om )
)
13 simprr 774 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  A  ~~  x )
1412, 13jca 541 . . . . . . . . 9  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
154, 14sylanr2 665 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  x  ~~  A ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
1615ex 441 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  ( ( x  e.  On  /\  x  ~~  A )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) ) )
1716reximdv2 2855 . . . . . 6  |-  ( -.  A  e.  Fin  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
1817com12 31 . . . . 5  |-  ( E. x  e.  On  x  ~~  A  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On 
\  om ) A 
~~  x ) )
193, 18sylbi 200 . . . 4  |-  ( A  e.  dom  card  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
2019con1d 129 . . 3  |-  ( A  e.  dom  card  ->  ( -.  E. x  e.  ( On  \  om ) A  ~~  x  ->  A  e.  Fin )
)
212, 20syl5com 30 . 2  |-  ( A  e. FinVII  ->  ( A  e. 
dom  card  ->  A  e.  Fin ) )
22 eldifi 3544 . . . . . . 7  |-  ( x  e.  ( On  \  om )  ->  x  e.  On )
23 ensym 7636 . . . . . . 7  |-  ( A 
~~  x  ->  x  ~~  A )
24 isnumi 8398 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
2522, 23, 24syl2an 485 . . . . . 6  |-  ( ( x  e.  ( On 
\  om )  /\  A  ~~  x )  ->  A  e.  dom  card )
2625rexlimiva 2868 . . . . 5  |-  ( E. x  e.  ( On 
\  om ) A 
~~  x  ->  A  e.  dom  card )
2726con3i 142 . . . 4  |-  ( -.  A  e.  dom  card  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
28 isfin7 8749 . . . 4  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. x  e.  ( On  \  om ) A  ~~  x ) )
2927, 28syl5ibr 229 . . 3  |-  ( A  e.  V  ->  ( -.  A  e.  dom  card 
->  A  e. FinVII ) )
30 fin17 8842 . . . 4  |-  ( A  e.  Fin  ->  A  e. FinVII )
3130a1i 11 . . 3  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  A  e. FinVII ) )
3229, 31jad 167 . 2  |-  ( A  e.  V  ->  (
( A  e.  dom  card 
->  A  e.  Fin )  ->  A  e. FinVII ) )
3321, 32impbid2 209 1  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  ( A  e.  dom  card 
->  A  e.  Fin ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    e. wcel 1904   E.wrex 2757    \ cdif 3387   class class class wbr 4395   dom cdm 4839   Oncon0 5430   omcom 6711    ~~ cen 7584   Fincfn 7587   cardccrd 8387  FinVIIcfin7 8732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-fin7 8739
This theorem is referenced by:  fin71num  8845  dffin7-2  8846
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