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Theorem ficardom 7848
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 7131 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 187 . 2  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
3 finnum 7835 . . . . . . . 8  |-  ( A  e.  Fin  ->  A  e.  dom  card )
4 cardid2 7840 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
53, 4syl 16 . . . . . . 7  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
6 entr 7159 . . . . . . 7  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
75, 6sylan 458 . . . . . 6  |-  ( ( A  e.  Fin  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
8 cardon 7831 . . . . . . 7  |-  ( card `  A )  e.  On
9 onomeneq 7296 . . . . . . 7  |-  ( ( ( card `  A
)  e.  On  /\  x  e.  om )  ->  ( ( card `  A
)  ~~  x  <->  ( card `  A )  =  x ) )
108, 9mpan 652 . . . . . 6  |-  ( x  e.  om  ->  (
( card `  A )  ~~  x  <->  ( card `  A
)  =  x ) )
117, 10syl5ib 211 . . . . 5  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  =  x ) )
12 eleq1a 2505 . . . . 5  |-  ( x  e.  om  ->  (
( card `  A )  =  x  ->  ( card `  A )  e.  om ) )
1311, 12syld 42 . . . 4  |-  ( x  e.  om  ->  (
( A  e.  Fin  /\  A  ~~  x )  ->  ( card `  A
)  e.  om )
)
1413exp3acom23 1381 . . 3  |-  ( x  e.  om  ->  ( A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) ) )
1514rexlimiv 2824 . 2  |-  ( E. x  e.  om  A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) )
162, 15mpcom 34 1  |-  ( A  e.  Fin  ->  ( card `  A )  e. 
om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   Oncon0 4581   omcom 4845   dom cdm 4878   ` cfv 5454    ~~ cen 7106   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  cardnn  7850  isinffi  7879  finnisoeu  7994  iunfictbso  7995  ficardun  8082  ficardun2  8083  pwsdompw  8084  ackbij1lem5  8104  ackbij1lem9  8108  ackbij1lem10  8109  ackbij1lem14  8113  ackbij1b  8119  ackbij2lem2  8120  ackbij2  8123  fin23lem22  8207  fin1a2lem11  8290  domtriomlem  8322  pwfseqlem4a  8536  pwfseqlem4  8537  hashkf  11620  hashginv  11622  hashcard  11638  hashcl  11639  hashdom  11653  hashun  11656  ackbijnn  12607  mreexexd  13873  ishashinf  24159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826
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