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Theorem ficard 8445
Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ficard  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )

Proof of Theorem ficard
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 7134 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 carden 8431 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  <->  A  ~~  x ) )
3 cardnn 7855 . . . . . . . 8  |-  ( x  e.  om  ->  ( card `  x )  =  x )
4 eqtr 2455 . . . . . . . . 9  |-  ( ( ( card `  A
)  =  ( card `  x )  /\  ( card `  x )  =  x )  ->  ( card `  A )  =  x )
54expcom 426 . . . . . . . 8  |-  ( (
card `  x )  =  x  ->  ( (
card `  A )  =  ( card `  x
)  ->  ( card `  A )  =  x ) )
63, 5syl 16 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  =  x ) )
7 eleq1a 2507 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
86, 7syld 43 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  e.  om ) )
98adantl 454 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  ->  ( card `  A )  e. 
om ) )
102, 9sylbird 228 . . . 4  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( A  ~~  x  ->  ( card `  A
)  e.  om )
)
1110rexlimdva 2832 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  om  A  ~~  x  ->  ( card `  A )  e. 
om ) )
121, 11syl5bi 210 . 2  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
13 cardnn 7855 . . . . . . . 8  |-  ( (
card `  A )  e.  om  ->  ( card `  ( card `  A
) )  =  (
card `  A )
)
1413eqcomd 2443 . . . . . . 7  |-  ( (
card `  A )  e.  om  ->  ( card `  A )  =  (
card `  ( card `  A ) ) )
1514adantl 454 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  ( card `  A )  =  ( card `  ( card `  A ) ) )
16 carden 8431 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  (
( card `  A )  =  ( card `  ( card `  A ) )  <-> 
A  ~~  ( card `  A ) ) )
1715, 16mpbid 203 . . . . 5  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  A  ~~  ( card `  A
) )
1817ex 425 . . . 4  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  ~~  ( card `  A )
) )
1918ancld 538 . . 3  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  ( ( card `  A )  e. 
om  /\  A  ~~  ( card `  A )
) ) )
20 breq2 4219 . . . . 5  |-  ( x  =  ( card `  A
)  ->  ( A  ~~  x  <->  A  ~~  ( card `  A ) ) )
2120rspcev 3054 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  E. x  e.  om  A  ~~  x
)
2221, 1sylibr 205 . . 3  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  A  e.  Fin )
2319, 22syl6 32 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  e.  Fin ) )
2412, 23impbid 185 1  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4215   omcom 4848   ` cfv 5457    ~~ cen 7109   Fincfn 7112   cardccrd 7827
This theorem is referenced by:  cfpwsdom  8464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-ac2 8348
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-ac 8002
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