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Theorem ficardom 8338
Description: The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
Assertion
Ref Expression
ficardom  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )

Proof of Theorem ficardom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 7536 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 194 . 2  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
3 finnum 8325 . . . . . . . 8  |-  ( A  e.  Fin  ->  A  e.  dom  card )
4 cardid2 8330 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
53, 4syl 16 . . . . . . 7  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
6 entr 7564 . . . . . . 7  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
75, 6sylan 471 . . . . . 6  |-  ( ( A  e.  Fin  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
8 cardon 8321 . . . . . . 7  |-  ( card `  A )  e.  On
9 onomeneq 7704 . . . . . . 7  |-  ( ( ( card `  A
)  e.  On  /\  x  e.  om )  ->  ( ( card `  A
)  ~~  x  <->  ( card `  A )  =  x ) )
108, 9mpan 670 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  ~~  x  <->  ( card `  A
)  =  x ) )
117, 10syl5ib 219 . . . . 5  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  =  x ) )
12 eleq1a 2550 . . . . 5  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
1311, 12syld 44 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  e.  om )
)
1413expcomd 438 . . 3  |-  ( x  e.  om  ->  ( A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) ) )
1514rexlimiv 2949 . 2  |-  ( E. x  e.  om  A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
162, 15mpcom 36 1  |-  ( 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 2815   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5586   omcom 6678    ~~ cen 7510   Fincfn 7513   cardccrd 8312
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316
This theorem is referenced by:  cardnn  8340  isinffi  8369  finnisoeu  8490  iunfictbso  8491  ficardun  8578  ficardun2  8579  pwsdompw  8580  ackbij1lem5  8600  ackbij1lem9  8604  ackbij1lem10  8605  ackbij1lem14  8609  ackbij1b  8615  ackbij2lem2  8616  ackbij2  8619  fin23lem22  8703  fin1a2lem11  8786  domtriomlem  8818  pwfseqlem4a  9035  pwfseqlem4  9036  hashkf  12371  hashginv  12373  hashcard  12391  hashcl  12392  hashdom  12411  hashun  12414  ackbijnn  13599  mreexexd  14899  ishashinf  27274
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