Theorem limsssuc 7050

Description: A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
limsssuc	⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))

Proof of Theorem limsssuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssucid 5802	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
2		sstr2 3610	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ suc 𝐵 → 𝐴 ⊆ suc 𝐵))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ suc 𝐵)
4		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	4	biimpcd 239	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐵 → 𝐵 ∈ 𝐴))
6		limsuc 7049	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
7	6	biimpa 501	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
8		limord 5784	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → Ord 𝐴)
9	8	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐴)
10		ordelord 5745	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
11	8, 10	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
12		ordsuc 7014	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
13	11, 12	sylib 208	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord suc 𝐵)
14		ordtri1 5756	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
15	9, 13, 14	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
16	15	con2bid 344	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (suc 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
17	7, 16	mpbid 222	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
18	17	ex 450	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
19	5, 18	sylan9r 690	. . . . . . . . . 10 ⊢ ((Lim 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
20	19	con2d 129	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
21	20	ex 450	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
22	21	com23 86	. . . . . . 7 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥 ∈ 𝐴 → ¬ 𝑥 = 𝐵)))
23	22	imp31 448	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐵)
24		ssel2 3598	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc 𝐵)
25		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
26	25	elsuc 5794	. . . . . . . . . 10 ⊢ (𝑥 ∈ suc 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
27	24, 26	sylib 208	. . . . . . . . 9 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
28	27	ord 392	. . . . . . . 8 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐵))
29	28	con1d 139	. . . . . . 7 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐵))
30	29	adantll 750	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐵))
31	23, 30	mpd 15	. . . . 5 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
32	31	ex 450	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
33	32	ssrdv 3609	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → 𝐴 ⊆ 𝐵)
34	33	ex 450	. 2 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
35	3, 34	impbid2 216	1 ⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  Ord word 5722  Lim wlim 5724  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  cardlim  8798