Theorem ficardom 8787

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.

Step	Hyp	Ref	Expression
1		isfi 7979	. . 3 |- ( A e. Fin <-> E. x e. om A ~~ x )
2	1	biimpi 206	. 2 |- ( A e. Fin -> E. x e. om A ~~ x )
3		finnum 8774	. . . . . . . 8 |- ( A e. Fin -> A e. dom card )
4		cardid2 8779	. . . . . . . 8 |- ( A e. dom card -> ( card ` A ) ~~ A )
5	3, 4	syl 17	. . . . . . 7 |- ( A e. Fin -> ( card ` A ) ~~ A )
6		entr 8008	. . . . . . 7 |- ( ( ( card ` A ) ~~ A /\ A ~~ x ) -> ( card ` A ) ~~ x )
7	5, 6	sylan 488	. . . . . 6 |- ( ( A e. Fin /\ A ~~ x ) -> ( card ` A ) ~~ x )
8		cardon 8770	. . . . . . 7 |- ( card ` A ) e. On
9		onomeneq 8150	. . . . . . 7 |- ( ( ( card ` A ) e. On /\ x e. om ) -> ( ( card ` A ) ~~ x <-> ( card ` A ) = x ) )
10	8, 9	mpan 706	. . . . . 6 |- ( x e. om -> ( ( card ` A ) ~~ x <-> ( card ` A ) = x ) )
11	7, 10	syl5ib 234	. . . . 5 |- ( x e. om -> ( ( A e. Fin /\ A ~~ x ) -> ( card ` A ) = x ) )
12		eleq1a 2696	. . . . 5 |- ( x e. om -> ( ( card ` A ) = x -> ( card ` A ) e. om ) )
13	11, 12	syld 47	. . . 4 |- ( x e. om -> ( ( A e. Fin /\ A ~~ x ) -> ( card ` A ) e. om ) )
14	13	expcomd 454	. . 3 |- ( x e. om -> ( A ~~ x -> ( A e. Fin -> ( card ` A ) e. om ) ) )
15	14	rexlimiv 3027	. 2 |- ( E. x e. om A ~~ x -> ( A e. Fin -> ( card ` A ) e. om ) )
16	2, 15	mpcom 38	1 |- ( A e. Fin -> ( card ` A ) e. om )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   ~~ cen 7952   Fincfn 7955   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  cardnn  8789  isinffi  8818  finnisoeu  8936  iunfictbso  8937  ficardun  9024  ficardun2  9025  pwsdompw  9026  ackbij1lem5  9046  ackbij1lem9  9050  ackbij1lem10  9051  ackbij1lem14  9055  ackbij1b  9061  ackbij2lem2  9062  ackbij2  9065  fin23lem22  9149  fin1a2lem11  9232  domtriomlem  9264  pwfseqlem4a  9483  pwfseqlem4  9484  hashkf  13119  hashginv  13121  hashcard  13146  hashcl  13147  hashdom  13168  hashun  13171  ishashinf  13247  ackbijnn  14560  mreexexd  16308  mreexexdOLD  16309