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Theorem isfin7-2 8765
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 8670 . . . 4  |-  ( A  e. FinVII  ->  ( A  e. FinVII  <->  -.  E. x  e.  ( On 
\  om ) A 
~~  x ) )
21ibi 241 . . 3  |-  ( A  e. FinVII  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
3 isnum2 8315 . . . . 5  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
4 ensym 7554 . . . . . . . . 9  |-  ( x 
~~  A  ->  A  ~~  x )
5 simprl 755 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  On )
6 enfi 7726 . . . . . . . . . . . . . . 15  |-  ( A 
~~  x  ->  ( A  e.  Fin  <->  x  e.  Fin ) )
7 onfin 7698 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  (
x  e.  Fin  <->  x  e.  om ) )
86, 7sylan9bbr 700 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( A  e.  Fin  <->  x  e.  om ) )
98biimprd 223 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( x  e.  om  ->  A  e.  Fin )
)
109con3d 133 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( -.  A  e. 
Fin  ->  -.  x  e.  om ) )
1110impcom 430 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  -.  x  e.  om )
125, 11eldifd 3480 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  ( On  \  om )
)
13 simprr 756 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  A  ~~  x )
1412, 13jca 532 . . . . . . . . 9  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
154, 14sylanr2 653 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  x  ~~  A ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
1615ex 434 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  ( ( x  e.  On  /\  x  ~~  A )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) ) )
1716reximdv2 2927 . . . . . 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 195 . . . 4  |-  ( A  e.  dom  card  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
2019con1d 124 . . 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 3619 . . . . . . 7  |-  ( x  e.  ( On  \  om )  ->  x  e.  On )
23 ensym 7554 . . . . . . 7  |-  ( A 
~~  x  ->  x  ~~  A )
24 isnumi 8316 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
2522, 23, 24syl2an 477 . . . . . 6  |-  ( ( x  e.  ( On 
\  om )  /\  A  ~~  x )  ->  A  e.  dom  card )
2625rexlimiva 2944 . . . . 5  |-  ( E. x  e.  ( On 
\  om ) A 
~~  x  ->  A  e.  dom  card )
2726con3i 135 . . . 4  |-  ( -.  A  e.  dom  card  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
28 isfin7 8670 . . . 4  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. x  e.  ( On  \  om ) A  ~~  x ) )
2927, 28syl5ibr 221 . . 3  |-  ( A  e.  V  ->  ( -.  A  e.  dom  card 
->  A  e. FinVII ) )
30 fin17 8763 . . . 4  |-  ( A  e.  Fin  ->  A  e. FinVII )
3130a1i 11 . . 3  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  A  e. FinVII ) )
3229, 31jad 162 . 2  |-  ( A  e.  V  ->  (
( A  e.  dom  card 
->  A  e.  Fin )  ->  A  e. FinVII ) )
3321, 32impbid2 204 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 184    /\ wa 369    e. wcel 1762   E.wrex 2808    \ cdif 3466   class class class wbr 4440   Oncon0 4871   dom cdm 4992   omcom 6671    ~~ cen 7503   Fincfn 7506   cardccrd 8305  FinVIIcfin7 8653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-fin7 8660
This theorem is referenced by:  fin71num  8766  dffin7-2  8767
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