Theorem tfindsg2 7847

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 6386	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onsucb 7801	. . 3 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 217	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ On)
4		eloni 6371	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
5		ordsucss 7802	. . . 4 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7	6	imp 407	. 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 234	. . . . 5 ⊢ (suc 𝐵 ∈ On → 𝜓)
14		eloni 6371	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → Ord 𝑦)
15		ordelsuc 7804	. . . . . . . . . 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 459	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
18		tfindsg2.6	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ 𝑦) → (𝜒 → 𝜃))
19	18	ex 413	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑦 → (𝜒 → 𝜃)))
21	17, 20	sylbird 259	. . . . . . 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 407	. . . . 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 413	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
26	25	adantr 481	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑)))
27		vex 3478	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
28		limelon 6425	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
29	27, 28	mpan 688	. . . . . . . . . 10 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
30		eloni 6371	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
31		ordelsuc 7804	. . . . . . . . . . . 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 6386	. . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 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 3175	. . . . . . . . . . . 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 459	. . . . . . . 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 231	. . . . . . 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 407	. . . . 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 7846	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵 ⊆ 𝐴) → 𝜏)
47	46	expl 458	. . 3 ⊢ (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
48	47	adantr 481	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵 ⊆ 𝐴) → 𝜏))
49	3, 7, 48	mp2and 697	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ⊆ wss 3947  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367

This theorem is referenced by:  oeordi  8583