Theorem tfindsg2 7873

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 6387	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onsucb 7828	. . 3 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 221	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ On)
4		eloni 6372	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
5		ordsucss 7829	. . . 4 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	4, 5	syl 18	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7	6	imp 412	. 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 6372	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → Ord 𝑦)
15		ordelsuc 7831	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
16	14, 15	sylan2 605	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
17	16	ancoms 464	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
18		tfindsg2.6	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))
19	18	ex 418	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
20	19	adantr 486	. . . . . . . 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 607	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
23	22	imp 412	. . . . 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 418	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
26	25	adantr 486	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
27		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
28		limelon 6428	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
29	27, 28	mpan 703	. . . . . . . . . 10 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
30		eloni 6372	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
31		ordelsuc 7831	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
32	30, 31	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
33		onelon 6387	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
34	33, 14	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
35	34, 15	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
36	35	anassrs 473	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
37	36	imbi1d 344	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → ((𝐵 ∈ 𝑦 → 𝜒) ↔ (suc 𝐵 ⊆ 𝑦 → 𝜒)))
38	37	ralbidva 3184	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) ↔ ∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒)))
39	38	imbi1d 344	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑) ↔ (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
40	32, 39	imbi12d 347	. . . . . . . . . 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 605	. . . . . . . . 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 464	. . . . . . . 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 607	. . . . . 6 ⊢ ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
45	44	imp 412	. . . . 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 7872	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝐴) → 𝜏)
47	46	expl 463	. . 3 ⊢ (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
48	47	adantr 486	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
49	3, 7, 48	mp2and 712	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368

This theorem is used by:  oeordi  8596