Theorem cardlim 4823

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 2073	. . . . . . . . . . 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 4610	. . . . . . . . . . . 12 |- ((x e. On /\ om (_ x) -> x ~~ suc x)
4	3	ex 373	. . . . . . . . . . 11 |- (x e. On -> (om (_ x -> x ~~ suc x))
5		limom 3136	. . . . . . . . . . . 12 |- Lim om
6		limsssuc 3111	. . . . . . . . . . . 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 469	. . . . . . . . 9 |- ((x e. On /\ (card` A) = suc x) -> (om (_ (card` A) -> x ~~ suc x))
10		breq2 2613	. . . . . . . . . 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 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 1804	. . . . . . . . . 10 |- x e. V
17	16	sucid 3041	. . . . . . . . 9 |- x e. suc x
18		eleq2 1527	. . . . . . . . 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 4821	. . . . . . . 8 |- (card` (card` A)) = (card` A)
21	19, 20	syl6eleqr 1551	. . . . . . 7 |- ((card` A) = suc x -> x e. (card` (card` A)))
22		cardne 4802	. . . . . . 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 1722	. . 3 |- (om (_ (card` A) -> -. E.x e. On (card` A) = suc x)
27		peano1 3139	. . . . . 6 |- (/) e. om
28		ssel 2053	. . . . . 6 |- (om (_ (card` A) -> ((/) e. om -> (/) e. (card` A)))
29	27, 28	mpi 44	. . . . 5 |- (om (_ (card` A) -> (/) e. (card` A))
30		n0i 2275	. . . . 5 |- ((/) e. (card` A) -> -. (card` A) = (/))
31		cardon 4799	. . . . . . . . 9 |- (card` A) e. On
32	31	onord 3085	. . . . . . . 8 |- Ord (card` A)
33		ordzsl 3106	. . . . . . . 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 776	. . . . . . 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 3127	. 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 772   = wceq 953   e. wcel 955  E.wrex 1638   (_ wss 2037  (/)c0 2270   class class class wbr 2609  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940  omcom 3121  ` cfv 3172   ~~ cen 4348  cardccrd 4785

This theorem is referenced by:  alephislim 4855

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-1o 4117  df-er 4245  df-en 4351  df-dom 4352  df-card 4788

Copyright terms: Public domain