MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin7-2 Structured version   Unicode version

Theorem isfin7-2 8277
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 8182 . . . 4  |-  ( A  e. FinVII  ->  ( A  e. FinVII  <->  -.  E. x  e.  ( On 
\  om ) A 
~~  x ) )
21ibi 234 . . 3  |-  ( A  e. FinVII  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
3 isnum2 7833 . . . . 5  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
4 ensym 7157 . . . . . . . . 9  |-  ( x 
~~  A  ->  A  ~~  x )
5 simprl 734 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  On )
6 enfi 7326 . . . . . . . . . . . . . . 15  |-  ( A 
~~  x  ->  ( A  e.  Fin  <->  x  e.  Fin ) )
7 onfin 7298 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  (
x  e.  Fin  <->  x  e.  om ) )
86, 7sylan9bbr 683 . . . . . . . . . . . . . 14  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( A  e.  Fin  <->  x  e.  om ) )
98biimprd 216 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( x  e.  om  ->  A  e.  Fin )
)
109con3d 128 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  A  ~~  x )  -> 
( -.  A  e. 
Fin  ->  -.  x  e.  om ) )
1110impcom 421 . . . . . . . . . . 11  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  -.  x  e.  om )
125, 11eldifd 3332 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  x  e.  ( On  \  om )
)
13 simprr 735 . . . . . . . . . 10  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  A  ~~  x )
1412, 13jca 520 . . . . . . . . 9  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  A  ~~  x ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
154, 14sylanr2 636 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  ( x  e.  On  /\  x  ~~  A ) )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) )
1615ex 425 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  ( ( x  e.  On  /\  x  ~~  A )  ->  ( x  e.  ( On  \  om )  /\  A  ~~  x
) ) )
1716reximdv2 2816 . . . . . 6  |-  ( -.  A  e.  Fin  ->  ( E. x  e.  On  x  ~~  A  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
1817com12 30 . . . . 5  |-  ( E. x  e.  On  x  ~~  A  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On 
\  om ) A 
~~  x ) )
193, 18sylbi 189 . . . 4  |-  ( A  e.  dom  card  ->  ( -.  A  e.  Fin  ->  E. x  e.  ( On  \  om ) A  ~~  x ) )
2019con1d 119 . . 3  |-  ( A  e.  dom  card  ->  ( -.  E. x  e.  ( On  \  om ) A  ~~  x  ->  A  e.  Fin )
)
212, 20syl5com 29 . 2  |-  ( A  e. FinVII  ->  ( A  e. 
dom  card  ->  A  e.  Fin ) )
22 eldifi 3470 . . . . . . 7  |-  ( x  e.  ( On  \  om )  ->  x  e.  On )
23 ensym 7157 . . . . . . 7  |-  ( A 
~~  x  ->  x  ~~  A )
24 isnumi 7834 . . . . . . 7  |-  ( ( x  e.  On  /\  x  ~~  A )  ->  A  e.  dom  card )
2522, 23, 24syl2an 465 . . . . . 6  |-  ( ( x  e.  ( On 
\  om )  /\  A  ~~  x )  ->  A  e.  dom  card )
2625rexlimiva 2826 . . . . 5  |-  ( E. x  e.  ( On 
\  om ) A 
~~  x  ->  A  e.  dom  card )
2726con3i 130 . . . 4  |-  ( -.  A  e.  dom  card  ->  -.  E. x  e.  ( On  \  om ) A  ~~  x )
28 isfin7 8182 . . . 4  |-  ( A  e.  V  ->  ( A  e. FinVII 
<->  -.  E. x  e.  ( On  \  om ) A  ~~  x ) )
2927, 28syl5ibr 214 . . 3  |-  ( A  e.  V  ->  ( -.  A  e.  dom  card 
->  A  e. FinVII ) )
30 fin17 8275 . . . 4  |-  ( A  e.  Fin  ->  A  e. FinVII )
3130a1i 11 . . 3  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  A  e. FinVII ) )
3229, 31jad 157 . 2  |-  ( A  e.  V  ->  (
( A  e.  dom  card 
->  A  e.  Fin )  ->  A  e. FinVII ) )
3321, 32impbid2 197 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 178    /\ wa 360    e. wcel 1726   E.wrex 2707    \ cdif 3318   class class class wbr 4213   Oncon0 4582   omcom 4846   dom cdm 4879    ~~ cen 7107   Fincfn 7110   cardccrd 7823  FinVIIcfin7 8165
This theorem is referenced by:  fin71num  8278  dffin7-2  8279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-fin7 8172
  Copyright terms: Public domain W3C validator