Theorem limsuclem 2360

Description: Lemma for limsuc 2361.

Hypotheses

Ref	Expression
on.1	⊢ A ∈ On
on.2	⊢ B ∈ On

Assertion

Ref	Expression
limsuclem	⊢ ((Lim A ∧ B ∈ A) → suc B ∈ A)

Proof of Theorem limsuclem

Step	Hyp	Ref	Expression
1		on.2	. . . . . . . 8 ⊢ B ∈ On
2		on.1	. . . . . . . 8 ⊢ A ∈ On
3	1, 2	onsucss 2359	. . . . . . 7 ⊢ (B ∈ A ↔ suc B ⊆ A)
4	1	onsuc 2353	. . . . . . . 8 ⊢ suc B ∈ On
5	4, 2	onssel 2357	. . . . . . 7 ⊢ (suc B ⊆ A ↔ (suc B ∈ A ∨ suc B = A))
6		df-or 197	. . . . . . 7 ⊢ ((suc B ∈ A ∨ suc B = A) ↔ (¬ suc B ∈ A → suc B = A))
7	3, 5, 6	3bitr 155	. . . . . 6 ⊢ (B ∈ A ↔ (¬ suc B ∈ A → suc B = A))
8	7	biimp 133	. . . . 5 ⊢ (B ∈ A → (¬ suc B ∈ A → suc B = A))
9		limuni 2284	. . . . . . . 8 ⊢ (Lim A → A = ∪A)
10		unieq 1927	. . . . . . . . 9 ⊢ (suc B = A → ∪suc B = ∪A)
11	1	onunisuc 2354	. . . . . . . . 9 ⊢ ∪suc B = B
12	10, 11	syl5reqr 1139	. . . . . . . 8 ⊢ (suc B = A → ∪A = B)
13	9, 12	sylan9eqr 1145	. . . . . . 7 ⊢ ((suc B = A ∧ Lim A) → A = B)
14		eqimss 1548	. . . . . . 7 ⊢ (A = B → A ⊆ B)
15	2	onssneli 2349	. . . . . . 7 ⊢ (A ⊆ B → ¬ B ∈ A)
16	13, 14, 15	3syl 21	. . . . . 6 ⊢ ((suc B = A ∧ Lim A) → ¬ B ∈ A)
17	16	exp 291	. . . . 5 ⊢ (suc B = A → (Lim A → ¬ B ∈ A))
18	8, 17	syl6 23	. . . 4 ⊢ (B ∈ A → (¬ suc B ∈ A → (Lim A → ¬ B ∈ A)))
19	18	com3r 35	. . 3 ⊢ (Lim A → (B ∈ A → (¬ suc B ∈ A → ¬ B ∈ A)))
20		ax-3 5	. . 3 ⊢ ((¬ suc B ∈ A → ¬ B ∈ A) → (B ∈ A → suc B ∈ A))
21	19, 20	syli 52	. 2 ⊢ (Lim A → (B ∈ A → suc B ∈ A))
22	21	imp 277	1 ⊢ ((Lim A ∧ B ∈ A) → suc B ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∪cuni 1919  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  limsuc 2361

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-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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