Theorem limsssuc 3934

Description: A class includes a limit ordinal iff the successor of the class includes it.

Assertion

Ref	Expression
limsssuc	|- (Lim A -> (A C_ B <-> A C_ suc B))

Proof of Theorem limsssuc

Step	Hyp	Ref	Expression
1		sssucid 3742	. . 3 |- B C_ suc B
2		sstr2 2623	. . 3 |- (A C_ B -> (B C_ suc B -> A C_ suc B))
3	1, 2	mpi 55	. 2 |- (A C_ B -> A C_ suc B)
4		eleq1 1957	. . . . . . . . . . . 12 |- (x = B -> (x e. A <-> B e. A))
5	4	biimpcd 172	. . . . . . . . . . 11 |- (x e. A -> (x = B -> B e. A))
6		limsuc 3933	. . . . . . . . . . . . . 14 |- (Lim A -> (B e. A <-> suc B e. A))
7	6	biimpa 460	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> suc B e. A)
8		limord 3723	. . . . . . . . . . . . . . . 16 |- (Lim A -> Ord A)
9	8	adantr 425	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord A)
10		ordelord 3680	. . . . . . . . . . . . . . . . 17 |- ((Ord A /\ B e. A) -> Ord B)
11	10, 8	sylan 497	. . . . . . . . . . . . . . . 16 |- ((Lim A /\ B e. A) -> Ord B)
12		ordsuc 3895	. . . . . . . . . . . . . . . 16 |- (Ord B <-> Ord suc B)
13	11, 12	sylib 215	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord suc B)
14		ordtri1 3693	. . . . . . . . . . . . . . 15 |- ((Ord A /\ Ord suc B) -> (A C_ suc B <-> -. suc B e. A))
15	9, 13, 14	syl11anc 524	. . . . . . . . . . . . . 14 |- ((Lim A /\ B e. A) -> (A C_ suc B <-> -. suc B e. A))
16	15	con2bid 585	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> (suc B e. A <-> -. A C_ suc B))
17	7, 16	mpbid 212	. . . . . . . . . . . 12 |- ((Lim A /\ B e. A) -> -. A C_ suc B)
18	17	ex 402	. . . . . . . . . . 11 |- (Lim A -> (B e. A -> -. A C_ suc B))
19	5, 18	sylan9r 519	. . . . . . . . . 10 |- ((Lim A /\ x e. A) -> (x = B -> -. A C_ suc B))
20	19	con2d 107	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> (A C_ suc B -> -. x = B))
21	20	ex 402	. . . . . . . 8 |- (Lim A -> (x e. A -> (A C_ suc B -> -. x = B)))
22	21	com23 36	. . . . . . 7 |- (Lim A -> (A C_ suc B -> (x e. A -> -. x = B)))
23	22	imp31 389	. . . . . 6 |- (((Lim A /\ A C_ suc B) /\ x e. A) -> -. x = B)
24		ssel2 2616	. . . . . . . . . 10 |- ((A C_ suc B /\ x e. A) -> x e. suc B)
25		visset 2295	. . . . . . . . . . 11 |- x e. _V
26	25	elsuc 3734	. . . . . . . . . 10 |- (x e. suc B <-> (x e. B \/ x = B))
27	24, 26	sylib 215	. . . . . . . . 9 |- ((A C_ suc B /\ x e. A) -> (x e. B \/ x = B))
28	27	ord 249	. . . . . . . 8 |- ((A C_ suc B /\ x e. A) -> (-. x e. B -> x = B))
29	28	con1d 109	. . . . . . 7 |- ((A C_ suc B /\ x e. A) -> (-. x = B -> x e. B))
30	29	adantll 428	. . . . . 6 |- (((Lim A /\ A C_ suc B) /\ x e. A) -> (-. x = B -> x e. B))
31	23, 30	mpd 29	. . . . 5 |- (((Lim A /\ A C_ suc B) /\ x e. A) -> x e. B)
32	31	ex 402	. . . 4 |- ((Lim A /\ A C_ suc B) -> (x e. A -> x e. B))
33	32	ssrdv 2622	. . 3 |- ((Lim A /\ A C_ suc B) -> A C_ B)
34	33	ex 402	. 2 |- (Lim A -> (A C_ suc B -> A C_ B))
35	3, 34	impbid2 576	1 |- (Lim A -> (A C_ B <-> A C_ suc B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  Ord word 3656  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  cardlim 6003

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663

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