Theorem limsssuc 7887

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 6475	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
2		sstr2 4015	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ suc 𝐵 → 𝐴 ⊆ suc 𝐵))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ suc 𝐵)
4		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	4	biimpcd 249	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐵 → 𝐵 ∈ 𝐴))
6		limsuc 7886	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
7	6	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
8		limord 6455	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → Ord 𝐴)
9		ordelord 6417	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
10	8, 9	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
11		ordsuc 7849	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
12	10, 11	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord suc 𝐵)
13		ordtri1 6428	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
14	8, 12, 13	syl2an2r 684	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
15	14	con2bid 354	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (suc 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
16	7, 15	mpbid 232	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
17	16	ex 412	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
18	5, 17	sylan9r 508	. . . . . . . . . 10 ⊢ ((Lim 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
19	18	con2d 134	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
20	19	ex 412	. . . . . . . 8 ⊢ (Lim 𝐴 → (𝑥 ∈ 𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
21	20	com23 86	. . . . . . 7 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥 ∈ 𝐴 → ¬ 𝑥 = 𝐵)))
22	21	imp31 417	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐵)
23		ssel2 4003	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc 𝐵)
24		vex 3492	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	elsuc 6465	. . . . . . . . . 10 ⊢ (𝑥 ∈ suc 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
26	23, 25	sylib 218	. . . . . . . . 9 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
27	26	ord 863	. . . . . . . 8 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝐵 → 𝑥 = 𝐵))
28	27	con1d 145	. . . . . . 7 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐵))
29	28	adantll 713	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐵))
30	22, 29	mpd 15	. . . . 5 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
31	30	ex 412	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
32	31	ssrdv 4014	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → 𝐴 ⊆ 𝐵)
33	32	ex 412	. 2 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
34	3, 33	impbid2 226	1 ⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  Ord word 6394  Lim wlim 6396  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401

This theorem is referenced by:  cardlim  10041