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Theorem ficard 8952
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 7551 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 carden 8938 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  om )  ->  ( ( card `  A
)  =  ( card `  x )  <->  A  ~~  x ) )
3 cardnn 8356 . . . . . . . 8  |-  ( x  e.  om  ->  ( card `  x )  =  x )
4 eqtr 2493 . . . . . . . . 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 2550 . . . . . . 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 2959 . . 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 8356 . . . . . . . 8  |-  ( (
card `  A )  e.  om  ->  ( card `  ( card `  A
) )  =  (
card `  A )
)
1413eqcomd 2475 . . . . . . 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 8938 . . . . . 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 4457 . . . . 5  |-  ( x  =  ( card `  A
)  ->  ( A  ~~  x  <->  A  ~~  ( card `  A ) ) )
2120rspcev 3219 . . . 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 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594   omcom 6695    ~~ cen 7525   Fincfn 7528   cardccrd 8328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-ac2 8855
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-om 6696  df-recs 7054  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-ac 8509
This theorem is referenced by:  cfpwsdom  8971
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