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Theorem ficard 8844
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 7446 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 carden 8830 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  <->  A  ~~  x ) )
3 cardnn 8248 . . . . . . . 8  |-  ( x  e.  om  ->  ( card `  x )  =  x )
4 eqtr 2480 . . . . . . . . 9  |-  ( ( ( card `  A
)  =  ( card `  x )  /\  ( card `  x )  =  x )  ->  ( card `  A )  =  x )
54expcom 435 . . . . . . . 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 2537 . . . . . . 7  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
86, 7syld 44 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  =  ( card `  x
)  ->  ( card `  A )  e.  om ) )
98adantl 466 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  ->  ( card `  A )  e. 
om ) )
102, 9sylbird 235 . . . 4  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( A  ~~  x  ->  ( card `  A
)  e.  om )
)
1110rexlimdva 2947 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  om  A  ~~  x  ->  ( card `  A )  e. 
om ) )
121, 11syl5bi 217 . 2  |-  ( A  e.  V  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
13 cardnn 8248 . . . . . . . 8  |-  ( (
card `  A )  e.  om  ->  ( card `  ( card `  A
) )  =  (
card `  A )
)
1413eqcomd 2462 . . . . . . 7  |-  ( (
card `  A )  e.  om  ->  ( card `  A )  =  (
card `  ( card `  A ) ) )
1514adantl 466 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  ( card `  A )  =  ( card `  ( card `  A ) ) )
16 carden 8830 . . . . . 6  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  (
( card `  A )  =  ( card `  ( card `  A ) )  <-> 
A  ~~  ( card `  A ) ) )
1715, 16mpbid 210 . . . . 5  |-  ( ( A  e.  V  /\  ( card `  A )  e.  om )  ->  A  ~~  ( card `  A
) )
1817ex 434 . . . 4  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  ~~  ( card `  A )
) )
1918ancld 553 . . 3  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  ( ( card `  A )  e. 
om  /\  A  ~~  ( card `  A )
) ) )
20 breq2 4407 . . . . 5  |-  ( x  =  ( card `  A
)  ->  ( A  ~~  x  <->  A  ~~  ( card `  A ) ) )
2120rspcev 3179 . . . 4  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  E. x  e.  om  A  ~~  x
)
2221, 1sylibr 212 . . 3  |-  ( ( ( card `  A
)  e.  om  /\  A  ~~  ( card `  A
) )  ->  A  e.  Fin )
2319, 22syl6 33 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  om  ->  A  e.  Fin ) )
2412, 23impbid 191 1  |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e.  om ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4403   ` cfv 5529   omcom 6589    ~~ cen 7420   Fincfn 7423   cardccrd 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-ac2 8747
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-om 6590  df-recs 6945  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-ac 8401
This theorem is referenced by:  cfpwsdom  8863
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