Theorem findsg 7590

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. The basis of this version is an arbitrary natural number 𝐵 instead of zero. (Contributed by NM, 16-Sep-1995.)

Hypotheses

Ref	Expression
findsg.1	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
findsg.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
findsg.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
findsg.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
findsg.5	⊢ (𝐵 ∈ ω → 𝜓)
findsg.6	⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))

Assertion

Ref	Expression
findsg	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → 𝜏)

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

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

Proof of Theorem findsg

Step	Hyp	Ref	Expression
1		sseq2 3941	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
2	1	adantl 485	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
3		eqeq2 2810	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝑥 = 𝐵 ↔ 𝑥 = ∅))
4		findsg.1	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
5	3, 4	syl6bir 257	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝑥 = ∅ → (𝜑 ↔ 𝜓)))
6	5	imp 410	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑 ↔ 𝜓))
7	2, 6	imbi12d 348	. . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
8	1	imbi1d 345	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9		ss0 4306	. . . . . . . . 9 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
10	9	con3i 157	. . . . . . . 8 ⊢ (¬ 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
11	10	pm2.21d 121	. . . . . . 7 ⊢ (¬ 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑 ↔ 𝜓)))
12	11	pm5.74d 276	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
13	8, 12	sylan9bbr 514	. . . . 5 ⊢ ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
14	7, 13	pm2.61ian 811	. . . 4 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
15	14	imbi2d 344	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))))
16		sseq2 3941	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
17		findsg.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
18	16, 17	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝑦 → 𝜒)))
19	18	imbi2d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ 𝑦 → 𝜒))))
20		sseq2 3941	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ suc 𝑦))
21		findsg.3	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
22	20, 21	imbi12d 348	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ suc 𝑦 → 𝜃)))
23	22	imbi2d 344	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦 → 𝜃))))
24		sseq2 3941	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
25		findsg.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
26	24, 25	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝐴 → 𝜏)))
27	26	imbi2d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ 𝐴 → 𝜏))))
28		findsg.5	. . . 4 ⊢ (𝐵 ∈ ω → 𝜓)
29	28	a1d 25	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))
30		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
31	30	sucex 7506	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ∈ V
32	31	eqvinc 3590	. . . . . . . . . . . 12 ⊢ (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵))
33	28, 4	syl5ibr 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ ω → 𝜑))
34	21	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝜑 → 𝜃))
35	33, 34	sylan9r 512	. . . . . . . . . . . . 13 ⊢ ((𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
36	35	exlimiv 1931	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
37	32, 36	sylbi 220	. . . . . . . . . . 11 ⊢ (suc 𝑦 = 𝐵 → (𝐵 ∈ ω → 𝜃))
38	37	eqcoms 2806	. . . . . . . . . 10 ⊢ (𝐵 = suc 𝑦 → (𝐵 ∈ ω → 𝜃))
39	38	imim2i 16	. . . . . . . . 9 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃)))
40	39	a1d 25	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃))))
41	40	com4r 94	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
42	41	adantl 485	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
43		df-ne 2988	. . . . . . . . 9 ⊢ (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
44	43	anbi2i 625	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45		annim 407	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦))
46	44, 45	bitri 278	. . . . . . 7 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦))
47		nnon 7566	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
48		nnon 7566	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
49		onsssuc 6246	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦))
50		suceloni 7508	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
51		onelpss 6199	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
52	50, 51	sylan2 595	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
53	49, 52	bitrd 282	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
54	47, 48, 53	syl2anr 599	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
55		findsg.6	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))
56	55	ex 416	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
57	56	a1ddd 80	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
58	57	a2d 29	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ 𝑦 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
59	58	com23 86	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
60	54, 59	sylbird 263	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
61	46, 60	syl5bir 246	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
62	42, 61	pm2.61d 182	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃)))
63	62	ex 416	. . . 4 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
64	63	a2d 29	. . 3 ⊢ (𝑦 ∈ ω → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑦 → 𝜒)) → (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦 → 𝜃))))
65	15, 19, 23, 27, 29, 64	finds 7589	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵 ⊆ 𝐴 → 𝜏)))
66	65	imp31 421	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ∅c0 4243  Oncon0 6159  suc csuc 6161  ωcom 7560

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-om 7561

This theorem is referenced by:  nnaordi  8227  inf3lem5  9079  ackbij2lem4  9653  sornom  9688  fin23lem15  9745  fin23lem36  9759  isf32lem1  9764  isf32lem2  9765  wunex2  10149  indpi  10318  satfsschain  32724