MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ficardom Structured version   Unicode version

Theorem ficardom 8373
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 7576 . . 3  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
21biimpi 194 . 2  |-  ( A  e.  Fin  ->  E. x  e.  om  A  ~~  x
)
3 finnum 8360 . . . . . . . 8  |-  ( A  e.  Fin  ->  A  e.  dom  card )
4 cardid2 8365 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
53, 4syl 17 . . . . . . 7  |-  ( A  e.  Fin  ->  ( card `  A )  ~~  A )
6 entr 7604 . . . . . . 7  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
75, 6sylan 469 . . . . . 6  |-  ( ( A  e.  Fin  /\  A  ~~  x )  -> 
( card `  A )  ~~  x )
8 cardon 8356 . . . . . . 7  |-  ( card `  A )  e.  On
9 onomeneq 7744 . . . . . . 7  |-  ( ( ( card `  A
)  e.  On  /\  x  e.  om )  ->  ( ( card `  A
)  ~~  x  <->  ( card `  A )  =  x ) )
108, 9mpan 668 . . . . . 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 2485 . . . . 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 )
)
1413expcomd 436 . . 3  |-  ( x  e.  om  ->  ( A  ~~  x  ->  ( A  e.  Fin  ->  ( card `  A )  e. 
om ) ) )
1514rexlimiv 2889 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   class class class wbr 4394   dom cdm 4822   Oncon0 5409   ` cfv 5568   omcom 6682    ~~ cen 7550   Fincfn 7553   cardccrd 8347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351
This theorem is referenced by:  cardnn  8375  isinffi  8404  finnisoeu  8525  iunfictbso  8526  ficardun  8613  ficardun2  8614  pwsdompw  8615  ackbij1lem5  8635  ackbij1lem9  8639  ackbij1lem10  8640  ackbij1lem14  8644  ackbij1b  8650  ackbij2lem2  8651  ackbij2  8654  fin23lem22  8738  fin1a2lem11  8821  domtriomlem  8853  pwfseqlem4a  9068  pwfseqlem4  9069  hashkf  12452  hashginv  12454  hashcard  12472  hashcl  12473  hashdom  12493  hashun  12496  ackbijnn  13789  mreexexd  15260  ishashinf  28043
  Copyright terms: Public domain W3C validator