Theorem limsuc 2361

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

Assertion

Ref	Expression
limsuc	⊢ (Lim A → (B ∈ A ↔ suc B ∈ A))

Proof of Theorem limsuc

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

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ 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

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