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 ∈ A ↔ B ∈ A))
6	5	biimpcd 137	. . . . . . . . . . 11 ⊢ (x ∈ A → (x = B → B ∈ A))
7		limsuc 2361	. . . . . . . . . . . . . 14 ⊢ (Lim A → (B ∈ A ↔ suc B ∈ A))
8	7	biimpa 324	. . . . . . . . . . . . 13 ⊢ ((Lim A ∧ B ∈ A) → suc B ∈ A)
9		ordtri1 2231	. . . . . . . . . . . . . . 15 ⊢ ((Ord A ∧ Ord suc B) → (A ⊆ suc B ↔ ¬ suc B ∈ A))
10		limord 2283	. . . . . . . . . . . . . . . 16 ⊢ (Lim A → Ord A)
11	10	adantr 306	. . . . . . . . . . . . . . 15 ⊢ ((Lim A ∧ B ∈ A) → Ord A)
12		ordelord 2221	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord A ∧ B ∈ A) → Ord B)
13	12, 10	sylan 343	. . . . . . . . . . . . . . . 16 ⊢ ((Lim A ∧ B ∈ A) → Ord B)
14		ordsuc 2318	. . . . . . . . . . . . . . . 16 ⊢ (Ord B ↔ Ord suc B)
15	13, 14	sylib 173	. . . . . . . . . . . . . . 15 ⊢ ((Lim A ∧ B ∈ A) → Ord suc B)
16	9, 11, 15	sylanc 361	. . . . . . . . . . . . . 14 ⊢ ((Lim A ∧ B ∈ A) → (A ⊆ suc B ↔ ¬ suc B ∈ A))
17	16	bicon2d 404	. . . . . . . . . . . . 13 ⊢ ((Lim A ∧ B ∈ A) → (suc B ∈ A ↔ ¬ A ⊆ suc B))
18	8, 17	mpbid 170	. . . . . . . . . . . 12 ⊢ ((Lim A ∧ B ∈ A) → ¬ A ⊆ suc B)
19	18	exp 291	. . . . . . . . . . 11 ⊢ (Lim A → (B ∈ A → ¬ A ⊆ suc B))
20	6, 19	sylan9r 360	. . . . . . . . . 10 ⊢ ((Lim A ∧ x ∈ A) → (x = B → ¬ A ⊆ suc B))
21	20	con2d 83	. . . . . . . . 9 ⊢ ((Lim A ∧ x ∈ A) → (A ⊆ suc B → ¬ x = B))
22	21	exp 291	. . . . . . . 8 ⊢ (Lim A → (x ∈ A → (A ⊆ suc B → ¬ x = B)))
23	22	com23 32	. . . . . . 7 ⊢ (Lim A → (A ⊆ suc B → (x ∈ A → ¬ x = B)))
24	23	imp31 280	. . . . . 6 ⊢ (((Lim A ∧ A ⊆ suc B) ∧ x ∈ A) → ¬ x = B)
25		ssel2 1503	. . . . . . . . . 10 ⊢ ((A ⊆ suc B ∧ x ∈ A) → x ∈ suc B)
26		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
27	26	elsuc 2292	. . . . . . . . . 10 ⊢ (x ∈ suc B ↔ (x ∈ B ∨ x = B))
28	25, 27	sylib 173	. . . . . . . . 9 ⊢ ((A ⊆ suc B ∧ x ∈ A) → (x ∈ B ∨ x = B))
29	28	ord 202	. . . . . . . 8 ⊢ ((A ⊆ suc B ∧ x ∈ A) → (¬ x ∈ B → x = B))
30	29	con1d 85	. . . . . . 7 ⊢ ((A ⊆ suc B ∧ x ∈ A) → (¬ x = B → x ∈ B))
31	30	adantll 309	. . . . . 6 ⊢ (((Lim A ∧ A ⊆ suc B) ∧ x ∈ A) → (¬ x = B → x ∈ B))
32	24, 31	mpd 46	. . . . 5 ⊢ (((Lim A ∧ A ⊆ suc B) ∧ x ∈ A) → x ∈ B)
33	32	exp 291	. . . 4 ⊢ ((Lim A ∧ A ⊆ suc B) → (x ∈ A → x ∈ 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   ∈ 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

metamath.org