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Theorem ficardom 7684
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 6973 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 186 . 2  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
3 finnum 7671 . . . . . . . 8  |-  ( A  e.  Fin  ->  A  e.  dom  card )
4 cardid2 7676 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
53, 4syl 15 . . . . . . 7  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
6 entr 7001 . . . . . . 7  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
75, 6sylan 457 . . . . . 6  |-  ( ( A  e.  Fin  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
8 cardon 7667 . . . . . . 7  |-  ( card `  A )  e.  On
9 onomeneq 7138 . . . . . . 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 2427 . . . . 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 1372 . . 3  |-  ( x  e.  om  ->  ( A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) ) )
1514rexlimiv 2737 . 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 1642    e. wcel 1710   E.wrex 2620   class class class wbr 4104   Oncon0 4474   omcom 4738   dom cdm 4771   ` cfv 5337    ~~ cen 6948   Fincfn 6951   cardccrd 7658
This theorem is referenced by:  cardnn  7686  isinffi  7715  finnisoeu  7830  iunfictbso  7831  ficardun  7918  ficardun2  7919  pwsdompw  7920  ackbij1lem5  7940  ackbij1lem9  7944  ackbij1lem10  7945  ackbij1lem14  7949  ackbij1b  7955  ackbij2lem2  7956  ackbij2  7959  fin23lem22  8043  fin1a2lem11  8126  domtriomlem  8158  pwfseqlem4a  8373  pwfseqlem4  8374  hashkf  11432  hashginv  11434  hashcard  11442  hashcl  11443  hashdom  11454  hashun  11457  ackbijnn  12383  mreexexd  13649  ishashinf  23363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662
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