Theorem limsuc 2361

Description: The successor of a member of a limit ordinal is also a member.

Assertion

Ref	Expression
limsuc	|- (Lim A -> (B e. A <-> suc B e. A))

Proof of Theorem limsuc

Step	Hyp	Ref	Expression
1		limord 2283	. . . 4 |- (Lim A -> Ord A)
2		ordeleqon 2241	. . . 4 |- (Ord A <-> (A e. On \/ A = On))
3	1, 2	sylib 173	. . 3 |- (Lim A -> (A e. On \/ A = On))
4		onelon 2223	. . . . . . 7 |- ((A e. On /\ B e. A) -> B e. On)
5		limeq 2211	. . . . . . . . . . . . . 14 |- (A = if(A e. On, A, (/)) -> (Lim A <-> Lim if(A e. On, A, (/))))
6		eleq2 1150	. . . . . . . . . . . . . 14 |- (A = if(A e. On, A, (/)) -> (B e. A <-> B e. if(A e. On, A, (/))))
7	5, 6	anbi12d 476	. . . . . . . . . . . . 13 |- (A = if(A e. On, A, (/)) -> ((Lim A /\ B e. A) <-> (Lim if(A e. On, A, (/)) /\ B e. if(A e. On, A, (/)))))
8		eleq2 1150	. . . . . . . . . . . . 13 |- (A = if(A e. On, A, (/)) -> (suc B e. A <-> suc B e. if(A e. On, A, (/))))
9	7, 8	imbi12d 474	. . . . . . . . . . . 12 |- (A = if(A e. On, A, (/)) -> (((Lim A /\ B e. A) -> suc B e. A) <-> ((Lim if(A e. On, A, (/)) /\ B e. if(A e. On, A, (/))) -> suc B e. if(A e. On, A, (/)))))
10		eleq1 1149	. . . . . . . . . . . . . 14 |- (B = if(B e. On, B, (/)) -> (B e. if(A e. On, A, (/)) <-> if(B e. On, B, (/)) e. if(A e. On, A, (/))))
11	10	anbi2d 468	. . . . . . . . . . . . 13 |- (B = if(B e. On, B, (/)) -> ((Lim if(A e. On, A, (/)) /\ B e. if(A e. On, A, (/))) <-> (Lim if(A e. On, A, (/)) /\ if(B e. On, B, (/)) e. if(A e. On, A, (/)))))
12		suceq 2288	. . . . . . . . . . . . . 14 |- (B = if(B e. On, B, (/)) -> suc B = suc if(B e. On, B, (/)))
13	12	eleq1d 1155	. . . . . . . . . . . . 13 |- (B = if(B e. On, B, (/)) -> (suc B e. if(A e. On, A, (/)) <-> suc if(B e. On, B, (/)) e. if(A e. On, A, (/))))
14	11, 13	imbi12d 474	. . . . . . . . . . . 12 |- (B = if(B e. On, B, (/)) -> (((Lim if(A e. On, A, (/)) /\ B e. if(A e. On, A, (/))) -> suc B e. if(A e. On, A, (/))) <-> ((Lim if(A e. On, A, (/)) /\ if(B e. On, B, (/)) e. if(A e. On, A, (/))) -> suc if(B e. On, B, (/)) e. if(A e. On, A, (/)))))
15		0elon 2277	. . . . . . . . . . . . . 14 |- (/) e. On
16	15	elimel 1793	. . . . . . . . . . . . 13 |- if(A e. On, A, (/)) e. On
17	15	elimel 1793	. . . . . . . . . . . . 13 |- if(B e. On, B, (/)) e. On
18	16, 17	limsuclem 2360	. . . . . . . . . . . 12 |- ((Lim if(A e. On, A, (/)) /\ if(B e. On, B, (/)) e. if(A e. On, A, (/))) -> suc if(B e. On, B, (/)) e. if(A e. On, A, (/)))
19	9, 14, 18	dedth2h 1787	. . . . . . . . . . 11 |- ((A e. On /\ B e. On) -> ((Lim A /\ B e. A) -> suc B e. A))
20	19	exp4b 296	. . . . . . . . . 10 |- (A e. On -> (B e. On -> (Lim A -> (B e. A -> suc B e. A))))
21	20	com34 36	. . . . . . . . 9 |- (A e. On -> (B e. On -> (B e. A -> (Lim A -> suc B e. A))))
22	21	com23 32	. . . . . . . 8 |- (A e. On -> (B e. A -> (B e. On -> (Lim A -> suc B e. A))))
23	22	imp 277	. . . . . . 7 |- ((A e. On /\ B e. A) -> (B e. On -> (Lim A -> suc B e. A)))
24	4, 23	mpd 46	. . . . . 6 |- ((A e. On /\ B e. A) -> (Lim A -> suc B e. A))
25	24	exp 291	. . . . 5 |- (A e. On -> (B e. A -> (Lim A -> suc B e. A)))
26	25	com23 32	. . . 4 |- (A e. On -> (Lim A -> (B e. A -> suc B e. A)))
27		suceloni 2314	. . . . . 6 |- (B e. On -> suc B e. On)
28		eleq2 1150	. . . . . . 7 |- (A = On -> (B e. A <-> B e. On))
29		eleq2 1150	. . . . . . 7 |- (A = On -> (suc B e. A <-> suc B e. On))
30	28, 29	imbi12d 474	. . . . . 6 |- (A = On -> ((B e. A -> suc B e. A) <-> (B e. On -> suc B e. On)))
31	27, 30	mpbiri 169	. . . . 5 |- (A = On -> (B e. A -> suc B e. A))
32	31	a1d 14	. . . 4 |- (A = On -> (Lim A -> (B e. A -> suc B e. A)))
33	26, 32	jaoi 275	. . 3 |- ((A e. On \/ A = On) -> (Lim A -> (B e. A -> suc B e. A)))
34	3, 33	mpcom 49	. 2 |- (Lim A -> (B e. A -> suc B e. A))
35		ordtr 2213	. . 3 |- (Ord A -> Tr A)
36		trsuc 2308	. . . 4 |- ((Tr A /\ suc B e. A) -> B e. A)
37	36	exp 291	. . 3 |- (Tr A -> (suc B e. A -> B e. A))
38	1, 35, 37	3syl 21	. 2 |- (Lim A -> (suc B e. A -> B e. A))
39	34, 38	impbid 397	1 |- (Lim A -> (B e. A <-> suc B e. A))

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  (/)c0 1707  ifcif 1776  Tr wtr 2041  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  limsssuc 2362  peano2b 2388  oaordi 3148  omordi 3164  limenpsi 3400  r1ord 3499  r1pwcl 3530  alephordi 3679  cflim 3704

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

metamath.org