Theorem tfindsg2 7838

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 6357	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onsucb 7792	. . 3 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 218	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ On)
4		eloni 6342	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
5		ordsucss 7793	. . . 4 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7	6	imp 406	. 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 235	. . . . 5 ⊢ (suc 𝐵 ∈ On → 𝜓)
14		eloni 6342	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → Ord 𝑦)
15		ordelsuc 7795	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
16	14, 15	sylan2 593	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
17	16	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
18		tfindsg2.6	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))
19	18	ex 412	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
21	17, 20	sylbird 260	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
22	2, 21	sylan2br 595	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
23	22	imp 406	. . . . 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 412	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
26	25	adantr 480	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
27		vex 3451	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
28		limelon 6397	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
29	27, 28	mpan 690	. . . . . . . . . 10 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
30		eloni 6342	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
31		ordelsuc 7795	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
32	30, 31	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
33		onelon 6357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
34	33, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
35	34, 15	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦 ∈ 𝑥)) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
36	35	anassrs 467	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
37	36	imbi1d 341	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → ((𝐵 ∈ 𝑦 → 𝜒) ↔ (suc 𝐵 ⊆ 𝑦 → 𝜒)))
38	37	ralbidva 3154	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) ↔ ∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒)))
39	38	imbi1d 341	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑) ↔ (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
40	32, 39	imbi12d 344	. . . . . . . . . 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 593	. . . . . . . . 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 458	. . . . . . . 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 232	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
44	2, 43	sylan2br 595	. . . . . 6 ⊢ ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
45	44	imp 406	. . . . 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 7837	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝐴) → 𝜏)
47	46	expl 457	. . 3 ⊢ (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
48	47	adantr 480	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338

This theorem is referenced by:  oeordi  8551