Theorem limsssuc 3111

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

Assertion

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

Proof of Theorem limsssuc

Step	Hyp	Ref	Expression
1		sssucid 3037	. . 3 |- B (_ suc B
2		sstr2 2061	. . 3 |- (A (_ B -> (B (_ suc B -> A (_ suc B))
3	1, 2	mpi 44	. 2 |- (A (_ B -> A (_ suc B)
4		eleq1 1526	. . . . . . . . . . . 12 |- (x = B -> (x e. A <-> B e. A))
5	4	biimpcd 155	. . . . . . . . . . 11 |- (x e. A -> (x = B -> B e. A))
6		limsuc 3110	. . . . . . . . . . . . . 14 |- (Lim A -> (B e. A <-> suc B e. A))
7	6	biimpa 416	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> suc B e. A)
8		ordtri1 2970	. . . . . . . . . . . . . . 15 |- ((Ord A /\ Ord suc B) -> (A (_ suc B <-> -. suc B e. A))
9		limord 3018	. . . . . . . . . . . . . . . 16 |- (Lim A -> Ord A)
10	9	adantr 389	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord A)
11		ordelord 2960	. . . . . . . . . . . . . . . . 17 |- ((Ord A /\ B e. A) -> Ord B)
12	11, 9	sylan 448	. . . . . . . . . . . . . . . 16 |- ((Lim A /\ B e. A) -> Ord B)
13		ordsuc 3055	. . . . . . . . . . . . . . . 16 |- (Ord B <-> Ord suc B)
14	12, 13	sylib 198	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord suc B)
15	8, 10, 14	sylanc 471	. . . . . . . . . . . . . 14 |- ((Lim A /\ B e. A) -> (A (_ suc B <-> -. suc B e. A))
16	15	con2bid 524	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> (suc B e. A <-> -. A (_ suc B))
17	7, 16	mpbid 195	. . . . . . . . . . . 12 |- ((Lim A /\ B e. A) -> -. A (_ suc B)
18	17	ex 373	. . . . . . . . . . 11 |- (Lim A -> (B e. A -> -. A (_ suc B))
19	5, 18	sylan9r 469	. . . . . . . . . 10 |- ((Lim A /\ x e. A) -> (x = B -> -. A (_ suc B))
20	19	con2d 91	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> (A (_ suc B -> -. x = B))
21	20	ex 373	. . . . . . . 8 |- (Lim A -> (x e. A -> (A (_ suc B -> -. x = B)))
22	21	com23 32	. . . . . . 7 |- (Lim A -> (A (_ suc B -> (x e. A -> -. x = B)))
23	22	imp31 362	. . . . . 6 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> -. x = B)
24		ssel2 2054	. . . . . . . . . 10 |- ((A (_ suc B /\ x e. A) -> x e. suc B)
25		visset 1804	. . . . . . . . . . 11 |- x e. V
26	25	elsuc 3028	. . . . . . . . . 10 |- (x e. suc B <-> (x e. B \/ x = B))
27	24, 26	sylib 198	. . . . . . . . 9 |- ((A (_ suc B /\ x e. A) -> (x e. B \/ x = B))
28	27	ord 232	. . . . . . . 8 |- ((A (_ suc B /\ x e. A) -> (-. x e. B -> x = B))
29	28	con1d 93	. . . . . . 7 |- ((A (_ suc B /\ x e. A) -> (-. x = B -> x e. B))
30	29	adantll 392	. . . . . 6 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> (-. x = B -> x e. B))
31	23, 30	mpd 26	. . . . 5 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> x e. B)
32	31	ex 373	. . . 4 |- ((Lim A /\ A (_ suc B) -> (x e. A -> x e. B))
33	32	ssrdv 2060	. . 3 |- ((Lim A /\ A (_ suc B) -> A (_ B)
34	33	ex 373	. 2 |- (Lim A -> (A (_ suc B -> A (_ B))
35	3, 34	impbid2 516	1 |- (Lim A -> (A (_ B <-> A (_ suc B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   (_ wss 2037  Ord word 2937  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  cardlim 4823

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-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-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

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-v 1803  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-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944

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