Theorem limsssuc 7839

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 6445	. . 3 ⊢ 𝐵 ⊆ suc 𝐵
2		sstr2 3990	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ suc 𝐵 → 𝐴 ⊆ suc 𝐵))
3	1, 2	mpi 20	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ suc 𝐵)
4		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	4	biimpcd 248	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐵 → 𝐵 ∈ 𝐴))
6		limsuc 7838	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
7	6	biimpa 478	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ 𝐴)
8		limord 6425	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → Ord 𝐴)
9		ordelord 6387	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
10	8, 9	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
11		ordsuc 7801	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
12	10, 11	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord suc 𝐵)
13		ordtri1 6398	. . . . . . . . . . . . . . 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 355	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → (suc 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
16	7, 15	mpbid 231	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
17	16	ex 414	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
18	5, 17	sylan9r 510	. . . . . . . . . 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 414	. . . . . . . 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 419	. . . . . 6 ⊢ (((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐵)
23		ssel2 3978	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ suc 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ suc 𝐵)
24		vex 3479	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
25	24	elsuc 6435	. . . . . . . . . 10 ⊢ (𝑥 ∈ suc 𝐵 ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 = 𝐵))
26	23, 25	sylib 217	. . . . . . . . 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 414	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
32	31	ssrdv 3989	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐴 ⊆ suc 𝐵) → 𝐴 ⊆ 𝐵)
33	32	ex 414	. 2 ⊢ (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵))
34	3, 33	impbid2 225	1 ⊢ (Lim 𝐴 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ⊆ wss 3949  Ord word 6364  Lim wlim 6366  suc csuc 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371

This theorem is referenced by:  cardlim  9967