Theorem cardlim 5001

Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91.

Assertion

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

Proof of Theorem cardlim

Step	Hyp	Ref	Expression
1		sseq2 2135	. . . . . . . . . . 11 |- ((card` A) = suc x -> (om (_ (card` A) <-> om (_ suc x))
2	1	biimpd 151	. . . . . . . . . 10 |- ((card` A) = suc x -> (om (_ (card` A) -> om (_ suc x))
3		infensuc 4784	. . . . . . . . . . . 12 |- ((x e. On /\ om (_ x) -> x ~~ suc x)
4	3	ex 371	. . . . . . . . . . 11 |- (x e. On -> (om (_ x -> x ~~ suc x))
5		limom 3233	. . . . . . . . . . . 12 |- Lim om
6		limsssuc 3204	. . . . . . . . . . . 12 |- (Lim om -> (om (_ x <-> om (_ suc x))
7	5, 6	ax-mp 7	. . . . . . . . . . 11 |- (om (_ x <-> om (_ suc x)
8	4, 7	syl5ibr 205	. . . . . . . . . 10 |- (x e. On -> (om (_ suc x -> x ~~ suc x))
9	2, 8	sylan9r 471	. . . . . . . . 9 |- ((x e. On /\ (card` A) = suc x) -> (om (_ (card` A) -> x ~~ suc x))
10		breq2 2696	. . . . . . . . . 10 |- ((card` A) = suc x -> (x ~~ (card` A) <-> x ~~ suc x))
11	10	adantl 388	. . . . . . . . 9 |- ((x e. On /\ (card` A) = suc x) -> (x ~~ (card` A) <-> x ~~ suc x))
12	9, 11	sylibrd 202	. . . . . . . 8 |- ((x e. On /\ (card` A) = suc x) -> (om (_ (card` A) -> x ~~ (card` A)))
13	12	ex 371	. . . . . . 7 |- (x e. On -> ((card` A) = suc x -> (om (_ (card` A) -> x ~~ (card` A))))
14	13	com3r 35	. . . . . 6 |- (om (_ (card` A) -> (x e. On -> ((card` A) = suc x -> x ~~ (card` A))))
15	14	imp 348	. . . . 5 |- ((om (_ (card` A) /\ x e. On) -> ((card` A) = suc x -> x ~~ (card` A)))
16		visset 1859	. . . . . . . . . 10 |- x e. V
17	16	sucid 3051	. . . . . . . . 9 |- x e. suc x
18		eleq2 1578	. . . . . . . . 9 |- ((card` A) = suc x -> (x e. (card` A) <-> x e. suc x))
19	17, 18	mpbiri 192	. . . . . . . 8 |- ((card` A) = suc x -> x e. (card` A))
20		cardidm 4999	. . . . . . . 8 |- (card` (card` A)) = (card` A)
21	19, 20	syl6eleqr 1602	. . . . . . 7 |- ((card` A) = suc x -> x e. (card` (card` A)))
22		cardne 4978	. . . . . . 7 |- (x e. (card` (card` A)) -> -. x ~~ (card` A))
23	21, 22	syl 10	. . . . . 6 |- ((card` A) = suc x -> -. x ~~ (card` A))
24	23	a1i 8	. . . . 5 |- ((om (_ (card` A) /\ x e. On) -> ((card` A) = suc x -> -. x ~~ (card` A)))
25	15, 24	pm2.65d 134	. . . 4 |- ((om (_ (card` A) /\ x e. On) -> -. (card` A) = suc x)
26	25	nrexdv 1776	. . 3 |- (om (_ (card` A) -> -. E.x e. On (card` A) = suc x)
27		peano1 3237	. . . . . 6 |- (/) e. om
28		ssel 2115	. . . . . 6 |- (om (_ (card` A) -> ((/) e. om -> (/) e. (card` A)))
29	27, 28	mpi 44	. . . . 5 |- (om (_ (card` A) -> (/) e. (card` A))
30		n0i 2337	. . . . 5 |- ((/) e. (card` A) -> -. (card` A) = (/))
31		cardon 4974	. . . . . . . . 9 |- (card` A) e. On
32	31	onordi 3074	. . . . . . . 8 |- Ord (card` A)
33		ordzsl 3199	. . . . . . . 8 |- (Ord (card` A) <-> ((card` A) = (/) \/ E.x e. On (card` A) = suc x \/ Lim (card` A)))
34	32, 33	mpbi 187	. . . . . . 7 |- ((card` A) = (/) \/ E.x e. On (card` A) = suc x \/ Lim (card` A))
35		3orass 784	. . . . . . 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 187	. . . . . 6 |- ((card` A) = (/) \/ (E.x e. On (card` A) = suc x \/ Lim (card` A)))
37	36	ori 228	. . . . 5 |- (-. (card` A) = (/) -> (E.x e. On (card` A) = suc x \/ Lim (card` A)))
38	29, 30, 37	3syl 20	. . . 4 |- (om (_ (card` A) -> (E.x e. On (card` A) = suc x \/ Lim (card` A)))
39	38	ord 230	. . 3 |- (om (_ (card` A) -> (-. E.x e. On (card` A) = suc x -> Lim (card` A)))
40	26, 39	mpd 26	. 2 |- (om (_ (card` A) -> Lim (card` A))
41		limomss 3224	. 2 |- (Lim (card` A) -> om (_ (card` A))
42	40, 41	impbii 155	1 |- (om (_ (card` A) <-> Lim (card` A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   \/ w3o 780   = wceq 992   e. wcel 994  E.wrex 1692   (_ wss 2099  (/)c0 2332   class class class wbr 2692  Ord word 2974  Oncon0 2975  Lim wlim 2976  suc csuc 2977  omcom 3218  ` cfv 3263   ~~ cen 4505  cardccrd 4959

This theorem is referenced by:  alephislim 5033

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-1o 4269  df-er 4401  df-en 4509  df-dom 4510  df-card 4962

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