Theorem limsssuc 7790

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 6392	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
2		sstr2 3922	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ suc 𝐵 → 𝐴 ⊆ suc 𝐵))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ suc 𝐵)
4		eleq1 2827	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	4	biimpcd 250	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐵 → 𝐵 ∈ 𝐴))
6		limsuc 7789	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
7	6	biimpa 477	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
8		limord 6371	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → Ord 𝐴)
9		ordelord 6332	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
10	8, 9	sylan 586	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
11		ordsuc 7754	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
12	10, 11	sylib 219	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord suc 𝐵)
13		ordtri1 6343	. . . . . . . . . . . . . . 15 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
14	8, 12, 13	syl2an2r 691	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ 𝐴))
15	14	con2bid 355	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (suc 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
16	7, 15	mpbid 233	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
17	16	ex 413	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
18	5, 17	sylan9r 513	. . . . . . . . . 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 413	. . . . . . . 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 418	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐵)
23		ssel2 3910	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc 𝐵)
24		vex 3435	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	elsuc 6382	. . . . . . . . . 10 ⊢ (𝑥 ∈ suc 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
26	23, 25	sylib 219	. . . . . . . . 9 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
27	26	ord 870	. . . . . . . 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 720	. . . . . 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 413	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
32	31	ssrdv 3921	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → 𝐴 ⊆ 𝐵)
33	32	ex 413	. 2 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
34	3, 33	impbid2 227	1 ⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  Ord word 6309  Lim wlim 6311  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316

This theorem is referenced by:  cardlim  9887