Theorem tfindsg2 7640

Description: Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal suc 𝐵 instead of zero. (Contributed by NM, 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
tfindsg2.1	⊢ (𝑥 = suc 𝐵 → (𝜑 ↔ 𝜓))
tfindsg2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
tfindsg2.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
tfindsg2.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
tfindsg2.5	⊢ (𝐵 ∈ On → 𝜓)
tfindsg2.6	⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))
tfindsg2.7	⊢ ((Lim 𝑥 ∧ 𝐵 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑))

Assertion

Ref	Expression
tfindsg2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfindsg2

Step	Hyp	Ref	Expression
1		onelon 6238	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		sucelon 7596	. . 3 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 221	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ On)
4		eloni 6223	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
5		ordsucss 7597	. . . 4 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7	6	imp 410	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ⊆ 𝐴)
8		tfindsg2.1	. . . . 5 ⊢ (𝑥 = suc 𝐵 → (𝜑 ↔ 𝜓))
9		tfindsg2.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
10		tfindsg2.3	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
11		tfindsg2.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
12		tfindsg2.5	. . . . . 6 ⊢ (𝐵 ∈ On → 𝜓)
13	2, 12	sylbir 238	. . . . 5 ⊢ (suc 𝐵 ∈ On → 𝜓)
14		eloni 6223	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → Ord 𝑦)
15		ordelsuc 7599	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
16	14, 15	sylan2 596	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
17	16	ancoms 462	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
18		tfindsg2.6	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))
19	18	ex 416	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
20	19	adantr 484	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
21	17, 20	sylbird 263	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
22	2, 21	sylan2br 598	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
23	22	imp 410	. . . . 5 ⊢ (((𝑦 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))
24		tfindsg2.7	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑))
25	24	ex 416	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
26	25	adantr 484	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
27		vex 3412	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
28		limelon 6276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
29	27, 28	mpan 690	. . . . . . . . . 10 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
30		eloni 6223	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
31		ordelsuc 7599	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
32	30, 31	sylan2 596	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
33		onelon 6238	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
34	33, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
35	34, 15	sylan2 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
36	35	anassrs 471	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
37	36	imbi1d 345	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → ((𝐵 ∈ 𝑦 → 𝜒) ↔ (suc 𝐵 ⊆ 𝑦 → 𝜒)))
38	37	ralbidva 3117	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) ↔ ∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒)))
39	38	imbi1d 345	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑) ↔ (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
40	32, 39	imbi12d 348	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)) ↔ (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))))
41	29, 40	sylan2 596	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)) ↔ (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))))
42	41	ancoms 462	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → ((𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)) ↔ (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))))
43	26, 42	mpbid 235	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
44	2, 43	sylan2br 598	. . . . . 6 ⊢ ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
45	44	imp 410	. . . . 5 ⊢ (((Lim 𝑥 ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))
46	8, 9, 10, 11, 13, 23, 45	tfindsg 7639	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝐴) → 𝜏)
47	46	expl 461	. . 3 ⊢ (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
48	47	adantr 484	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
49	3, 7, 48	mp2and 699	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ⊆ wss 3866  Ord word 6212  Oncon0 6213  Lim wlim 6214  suc csuc 6215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219

This theorem is referenced by:  oeordi  8315