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Theorem ficardom 7594
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 6885 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 186 . 2  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
3 finnum 7581 . . . . . . . 8  |-  ( A  e.  Fin  ->  A  e.  dom  card )
4 cardid2 7586 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
53, 4syl 15 . . . . . . 7  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
6 entr 6913 . . . . . . 7  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
75, 6sylan 457 . . . . . 6  |-  ( ( A  e.  Fin  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
8 cardon 7577 . . . . . . 7  |-  ( card `  A )  e.  On
9 onomeneq 7050 . . . . . . 7  |-  ( ( ( card `  A
)  e.  On  /\  x  e.  om )  ->  ( ( card `  A
)  ~~  x  <->  ( card `  A )  =  x ) )
108, 9mpan 651 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  ~~  x  <->  ( card `  A
)  =  x ) )
117, 10syl5ib 210 . . . . 5  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  =  x ) )
12 eleq1a 2352 . . . . 5  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
1311, 12syld 40 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  e.  om )
)
1413exp3acom23 1362 . . 3  |-  ( x  e.  om  ->  ( A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) ) )
1514rexlimiv 2661 . 2  |-  ( E. x  e.  om  A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
162, 15mpcom 32 1  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255    ~~ cen 6860   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  cardnn  7596  isinffi  7625  finnisoeu  7740  iunfictbso  7741  ficardun  7828  ficardun2  7829  pwsdompw  7830  ackbij1lem5  7850  ackbij1lem9  7854  ackbij1lem10  7855  ackbij1lem14  7859  ackbij1b  7865  ackbij2lem2  7866  ackbij2  7869  fin23lem22  7953  fin1a2lem11  8036  domtriomlem  8068  pwfseqlem4a  8283  pwfseqlem4  8284  hashkf  11339  hashginv  11341  hashcard  11349  hashcl  11350  hashdom  11361  hashun  11364  ackbijnn  12286  mreexexd  13550  ishashinf  23389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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