Theorem cardlim 8798

Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Assertion

Ref	Expression
cardlim	|- ( om C_ ( card ` A ) <-> Lim ( card ` A ) )

Proof of Theorem cardlim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3627	. . . . . . . . . . 11 |- ( ( card ` A ) = suc x -> ( om C_ ( card ` A ) <-> om C_ suc x ) )
2	1	biimpd 219	. . . . . . . . . 10 |- ( ( card ` A ) = suc x -> ( om C_ ( card ` A ) -> om C_ suc x ) )
3		limom 7080	. . . . . . . . . . . 12 |- Lim om
4		limsssuc 7050	. . . . . . . . . . . 12 |- ( Lim om -> ( om C_ x <-> om C_ suc x ) )
5	3, 4	ax-mp 5	. . . . . . . . . . 11 |- ( om C_ x <-> om C_ suc x )
6		infensuc 8138	. . . . . . . . . . . 12 |- ( ( x e. On /\ om C_ x ) -> x ~~ suc x )
7	6	ex 450	. . . . . . . . . . 11 |- ( x e. On -> ( om C_ x -> x ~~ suc x ) )
8	5, 7	syl5bir 233	. . . . . . . . . 10 |- ( x e. On -> ( om C_ suc x -> x ~~ suc x ) )
9	2, 8	sylan9r 690	. . . . . . . . 9 |- ( ( x e. On /\ ( card ` A ) = suc x ) -> ( om C_ ( card ` A ) -> x ~~ suc x ) )
10		breq2 4657	. . . . . . . . . 10 |- ( ( card ` A ) = suc x -> ( x ~~ ( card ` A ) <-> x ~~ suc x ) )
11	10	adantl 482	. . . . . . . . 9 |- ( ( x e. On /\ ( card ` A ) = suc x ) -> ( x ~~ ( card ` A ) <-> x ~~ suc x ) )
12	9, 11	sylibrd 249	. . . . . . . 8 |- ( ( x e. On /\ ( card ` A ) = suc x ) -> ( om C_ ( card ` A ) -> x ~~ ( card ` A ) ) )
13	12	ex 450	. . . . . . 7 |- ( x e. On -> ( ( card ` A ) = suc x -> ( om C_ ( card ` A ) -> x ~~ ( card ` A ) ) ) )
14	13	com3r 87	. . . . . 6 |- ( om C_ ( card ` A ) -> ( x e. On -> ( ( card ` A ) = suc x -> x ~~ ( card ` A ) ) ) )
15	14	imp 445	. . . . 5 |- ( ( om C_ ( card ` A ) /\ x e. On ) -> ( ( card ` A ) = suc x -> x ~~ ( card ` A ) ) )
16		vex 3203	. . . . . . . . . 10 |- x e. _V
17	16	sucid 5804	. . . . . . . . 9 |- x e. suc x
18		eleq2 2690	. . . . . . . . 9 |- ( ( card ` A ) = suc x -> ( x e. ( card ` A ) <-> x e. suc x ) )
19	17, 18	mpbiri 248	. . . . . . . 8 |- ( ( card ` A ) = suc x -> x e. ( card ` A ) )
20		cardidm 8785	. . . . . . . 8 |- ( card ` ( card ` A ) ) = ( card ` A )
21	19, 20	syl6eleqr 2712	. . . . . . 7 |- ( ( card ` A ) = suc x -> x e. ( card ` ( card ` A ) ) )
22		cardne 8791	. . . . . . 7 |- ( x e. ( card ` ( card ` A ) ) -> -. x ~~ ( card ` A ) )
23	21, 22	syl 17	. . . . . 6 |- ( ( card ` A ) = suc x -> -. x ~~ ( card ` A ) )
24	23	a1i 11	. . . . 5 |- ( ( om C_ ( card ` A ) /\ x e. On ) -> ( ( card ` A ) = suc x -> -. x ~~ ( card ` A ) ) )
25	15, 24	pm2.65d 187	. . . 4 |- ( ( om C_ ( card ` A ) /\ x e. On ) -> -. ( card ` A ) = suc x )
26	25	nrexdv 3001	. . 3 |- ( om C_ ( card ` A ) -> -. E. x e. On ( card ` A ) = suc x )
27		peano1 7085	. . . . . 6 |- (/) e. om
28		ssel 3597	. . . . . 6 |- ( om C_ ( card ` A ) -> ( (/) e. om -> (/) e. ( card ` A ) ) )
29	27, 28	mpi 20	. . . . 5 |- ( om C_ ( card ` A ) -> (/) e. ( card ` A ) )
30		n0i 3920	. . . . 5 |- ( (/) e. ( card ` A ) -> -. ( card ` A ) = (/) )
31		cardon 8770	. . . . . . . . 9 |- ( card ` A ) e. On
32	31	onordi 5832	. . . . . . . 8 |- Ord ( card ` A )
33		ordzsl 7045	. . . . . . . 8 |- ( Ord ( card ` A ) <-> ( ( card ` A ) = (/) \/ E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) )
34	32, 33	mpbi 220	. . . . . . 7 |- ( ( card ` A ) = (/) \/ E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) )
35		3orass 1040	. . . . . . 7 |- ( ( ( card ` A ) = (/) \/ E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) <-> ( ( card ` A ) = (/) \/ ( E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) ) )
36	34, 35	mpbi 220	. . . . . 6 |- ( ( card ` A ) = (/) \/ ( E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) )
37	36	ori 390	. . . . 5 |- ( -. ( card ` A ) = (/) -> ( E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) )
38	29, 30, 37	3syl 18	. . . 4 |- ( om C_ ( card ` A ) -> ( E. x e. On ( card ` A ) = suc x \/ Lim ( card ` A ) ) )
39	38	ord 392	. . 3 |- ( om C_ ( card ` A ) -> ( -. E. x e. On ( card ` A ) = suc x -> Lim ( card ` A ) ) )
40	26, 39	mpd 15	. 2 |- ( om C_ ( card ` A ) -> Lim ( card ` A ) )
41		limomss 7070	. 2 |- ( Lim ( card ` A ) -> om C_ ( card ` A ) )
42	40, 41	impbii 199	1 |- ( om C_ ( card ` A ) <-> Lim ( card ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ` cfv 5888   omcom 7065   ~~ cen 7952   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-csb 3534  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-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-card 8765

This theorem is referenced by:  infxpenlem  8836  alephislim  8906  cflim2  9085  winalim  9517  gruina  9640