Theorem limsssuc 2362

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 2300	. . . 4 |- B (_ suc B
2		sstr2 1510	. . . 4 |- (A (_ B -> (B (_ suc B -> A (_ suc B))
3	1, 2	mpi 44	. . 3 |- (A (_ B -> A (_ suc B)
4	3	a1i 7	. 2 |- (Lim A -> (A (_ B -> A (_ suc B))
5		eleq1 1149	. . . . . . . . . . . 12 |- (x = B -> (x e. A <-> B e. A))
6	5	biimpcd 137	. . . . . . . . . . 11 |- (x e. A -> (x = B -> B e. A))
7		limsuc 2361	. . . . . . . . . . . . . 14 |- (Lim A -> (B e. A <-> suc B e. A))
8	7	biimpa 324	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> suc B e. A)
9		ordtri1 2231	. . . . . . . . . . . . . . 15 |- ((Ord A /\ Ord suc B) -> (A (_ suc B <-> -. suc B e. A))
10		limord 2283	. . . . . . . . . . . . . . . 16 |- (Lim A -> Ord A)
11	10	adantr 306	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord A)
12		ordelord 2221	. . . . . . . . . . . . . . . . 17 |- ((Ord A /\ B e. A) -> Ord B)
13	12, 10	sylan 343	. . . . . . . . . . . . . . . 16 |- ((Lim A /\ B e. A) -> Ord B)
14		ordsuc 2318	. . . . . . . . . . . . . . . 16 |- (Ord B <-> Ord suc B)
15	13, 14	sylib 173	. . . . . . . . . . . . . . 15 |- ((Lim A /\ B e. A) -> Ord suc B)
16	9, 11, 15	sylanc 361	. . . . . . . . . . . . . 14 |- ((Lim A /\ B e. A) -> (A (_ suc B <-> -. suc B e. A))
17	16	bicon2d 404	. . . . . . . . . . . . 13 |- ((Lim A /\ B e. A) -> (suc B e. A <-> -. A (_ suc B))
18	8, 17	mpbid 170	. . . . . . . . . . . 12 |- ((Lim A /\ B e. A) -> -. A (_ suc B)
19	18	exp 291	. . . . . . . . . . 11 |- (Lim A -> (B e. A -> -. A (_ suc B))
20	6, 19	sylan9r 360	. . . . . . . . . 10 |- ((Lim A /\ x e. A) -> (x = B -> -. A (_ suc B))
21	20	con2d 83	. . . . . . . . 9 |- ((Lim A /\ x e. A) -> (A (_ suc B -> -. x = B))
22	21	exp 291	. . . . . . . 8 |- (Lim A -> (x e. A -> (A (_ suc B -> -. x = B)))
23	22	com23 32	. . . . . . 7 |- (Lim A -> (A (_ suc B -> (x e. A -> -. x = B)))
24	23	imp31 280	. . . . . 6 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> -. x = B)
25		ssel2 1503	. . . . . . . . . 10 |- ((A (_ suc B /\ x e. A) -> x e. suc B)
26		visset 1350	. . . . . . . . . . 11 |- x e. V
27	26	elsuc 2292	. . . . . . . . . 10 |- (x e. suc B <-> (x e. B \/ x = B))
28	25, 27	sylib 173	. . . . . . . . 9 |- ((A (_ suc B /\ x e. A) -> (x e. B \/ x = B))
29	28	ord 202	. . . . . . . 8 |- ((A (_ suc B /\ x e. A) -> (-. x e. B -> x = B))
30	29	con1d 85	. . . . . . 7 |- ((A (_ suc B /\ x e. A) -> (-. x = B -> x e. B))
31	30	adantll 309	. . . . . 6 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> (-. x = B -> x e. B))
32	24, 31	mpd 46	. . . . 5 |- (((Lim A /\ A (_ suc B) /\ x e. A) -> x e. B)
33	32	exp 291	. . . 4 |- ((Lim A /\ A (_ suc B) -> (x e. A -> x e. B))
34	33	ssrdv 1509	. . 3 |- ((Lim A /\ A (_ suc B) -> A (_ B)
35	34	exp 291	. 2 |- (Lim A -> (A (_ suc B -> A (_ B))
36	4, 35	impbid 397	1 |- (Lim A -> (A (_ B <-> A (_ suc B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  cardlim 3657

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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