Theorem cardlim 4862

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 2086	. . . . . . . . . . 11 |- ((card` A) = suc x -> (om (_ (card` A) <-> om (_ suc x))
2	1	biimpd 153	. . . . . . . . . 10 |- ((card` A) = suc x -> (om (_ (card` A) -> om (_ suc x))
3		infensuc 4648	. . . . . . . . . . . 12 |- ((x e. On /\ om (_ x) -> x ~~ suc x)
4	3	ex 373	. . . . . . . . . . 11 |- (x e. On -> (om (_ x -> x ~~ suc x))
5		limom 3152	. . . . . . . . . . . 12 |- Lim om
6		limsssuc 3127	. . . . . . . . . . . 12 |- (Lim om -> (om (_ x <-> om (_ suc x))
7	5, 6	ax-mp 7	. . . . . . . . . . 11 |- (om (_ x <-> om (_ suc x)
8	4, 7	syl5ibr 207	. . . . . . . . . 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 2628	. . . . . . . . . 10 |- ((card` A) = suc x -> (x ~~ (card` A) <-> x ~~ suc x))
11	10	adantl 390	. . . . . . . . 9 |- ((x e. On /\ (card` A) = suc x) -> (x ~~ (card` A) <-> x ~~ suc x))
12	9, 11	sylibrd 204	. . . . . . . 8 |- ((x e. On /\ (card` A) = suc x) -> (om (_ (card` A) -> x ~~ (card` A)))
13	12	ex 373	. . . . . . 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 350	. . . . 5 |- ((om (_ (card` A) /\ x e. On) -> ((card` A) = suc x -> x ~~ (card` A)))
16		visset 1816	. . . . . . . . . 10 |- x e. V
17	16	sucid 3057	. . . . . . . . 9 |- x e. suc x
18		eleq2 1538	. . . . . . . . 9 |- ((card` A) = suc x -> (x e. (card` A) <-> x e. suc x))
19	17, 18	mpbiri 194	. . . . . . . 8 |- ((card` A) = suc x -> x e. (card` A))
20		cardidm 4860	. . . . . . . 8 |- (card` (card` A)) = (card` A)
21	19, 20	syl6eleqr 1562	. . . . . . 7 |- ((card` A) = suc x -> x e. (card` (card` A)))
22		cardne 4840	. . . . . . 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 136	. . . 4 |- ((om (_ (card` A) /\ x e. On) -> -. (card` A) = suc x)
26	25	nrexdv 1733	. . 3 |- (om (_ (card` A) -> -. E.x e. On (card` A) = suc x)
27		peano1 3155	. . . . . 6 |- (/) e. om
28		ssel 2066	. . . . . 6 |- (om (_ (card` A) -> ((/) e. om -> (/) e. (card` A)))
29	27, 28	mpi 44	. . . . 5 |- (om (_ (card` A) -> (/) e. (card` A))
30		n0i 2288	. . . . 5 |- ((/) e. (card` A) -> -. (card` A) = (/))
31		cardon 4837	. . . . . . . . 9 |- (card` A) e. On
32	31	onord 3101	. . . . . . . 8 |- Ord (card` A)
33		ordzsl 3122	. . . . . . . 8 |- (Ord (card` A) <-> ((card` A) = (/) \/ E.x e. On (card` A) = suc x \/ Lim (card` A)))
34	32, 33	mpbi 189	. . . . . . 7 |- ((card` A) = (/) \/ E.x e. On (card` A) = suc x \/ Lim (card` A))
35		3orass 780	. . . . . . 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 189	. . . . . 6 |- ((card` A) = (/) \/ (E.x e. On (card` A) = suc x \/ Lim (card` A)))
37	36	ori 230	. . . . 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 232	. . 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 3143	. 2 |- (Lim (card` A) -> om (_ (card` A))
42	40, 41	impbi 157	1 |- (om (_ (card` A) <-> Lim (card` A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960  E.wrex 1649   (_ wss 2050  (/)c0 2283   class class class wbr 2624  Ord word 2953  Oncon0 2954  Lim wlim 2955  suc csuc 2956  omcom 3137  ` cfv 3188   ~~ cen 4370  cardccrd 4823

This theorem is referenced by:  alephislim 4894

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-1o 4139  df-er 4267  df-en 4374  df-dom 4375  df-card 4826

Copyright terms: Public domain