Theorem tfindsg2 7842

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 6371	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
2		onsucb 7797	. . 3 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
3	1, 2	sylib 220	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → suc 𝐵 ∈ On)
4		eloni 6356	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
5		ordsucss 7798	. . . 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 237	. . . . 5 ⊢ (suc 𝐵 ∈ On → 𝜓)
14		eloni 6356	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → Ord 𝑦)
15		ordelsuc 7800	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵 ∈ 𝑦 ↔ suc 𝐵 ⊆ 𝑦))
16	14, 15	sylan2 602	. . . . . . . . 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 262	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
22	2, 21	sylan2br 604	. . . . . 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 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
28		limelon 6411	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
29	27, 28	mpan 700	. . . . . . . . . 10 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
30		eloni 6356	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
31		ordelsuc 7800	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
32	30, 31	sylan2 602	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 ∈ 𝑥 ↔ suc 𝐵 ⊆ 𝑥))
33		onelon 6371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
34	33, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
35	34, 15	sylan2 602	. . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦 ∈ 𝑥) → ((𝐵 ∈ 𝑦 → 𝜒) ↔ (suc 𝐵 ⊆ 𝑦 → 𝜒)))
38	37	ralbidva 3183	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) ↔ ∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒)))
39	38	imbi1d 343	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦 ∈ 𝑥 (𝐵 ∈ 𝑦 → 𝜒) → 𝜑) ↔ (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
40	32, 39	imbi12d 346	. . . . . . . . . 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 602	. . . . . . . . 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 234	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ 𝐵 ∈ On) → (suc 𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (suc 𝐵 ⊆ 𝑦 → 𝜒) → 𝜑)))
44	2, 43	sylan2br 604	. . . . . 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 7841	. . . 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 709	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  Ord word 6345  Oncon0 6346  Lim wlim 6347  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352

This theorem is referenced by:  oeordi  8557