Theorem findsg 7903

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 3966	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
2	1	adantl 487	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
3		eqeq2 2778	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝑥 = 𝐵 ↔ 𝑥 = ∅))
4		findsg.1	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
5	3, 4	biimtrrdi 257	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝑥 = ∅ → (𝜑 ↔ 𝜓)))
6	5	imp 412	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑 ↔ 𝜓))
7	2, 6	imbi12d 347	. . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
8	1	imbi1d 344	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9		ss0 4362	. . . . . . . . 9 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
10	9	con3i 155	. . . . . . . 8 ⊢ (¬ 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
11	10	pm2.21d 122	. . . . . . 7 ⊢ (¬ 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑 ↔ 𝜓)))
12	11	pm5.74d 276	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
13	8, 12	sylan9bbr 520	. . . . 5 ⊢ ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
14	7, 13	pm2.61ian 824	. . . 4 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
15	14	imbi2d 343	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))))
16		sseq2 3966	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
17		findsg.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
18	16, 17	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝑦 → 𝜒)))
19	18	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ 𝑦 → 𝜒))))
20		sseq2 3966	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ suc 𝑦))
21		findsg.3	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
22	20, 21	imbi12d 347	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ suc 𝑦 → 𝜃)))
23	22	imbi2d 343	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦 → 𝜃))))
24		sseq2 3966	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
25		findsg.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
26	24, 25	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝐴 → 𝜏)))
27	26	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ 𝐴 → 𝜏))))
28		findsg.5	. . . 4 ⊢ (𝐵 ∈ ω → 𝜓)
29	28	a1d 26	. . 3 ⊢ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))
30		vex 3462	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
31	30	sucex 7814	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ∈ V
32	31	eqvinc 3611	. . . . . . . . . . . 12 ⊢ (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵))
33	28, 4	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ ω → 𝜑))
34	21	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝜑 → 𝜃))
35	33, 34	sylan9r 518	. . . . . . . . . . . . 13 ⊢ ((𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
36	35	exlimiv 1963	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (𝐵 = suc 𝑦 → (𝐵 ∈ ω → 𝜃))
39	38	imim2i 17	. . . . . . . . 9 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃)))
40	39	a1d 26	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃))))
41	40	com4r 95	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
42	41	adantl 487	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
43		df-ne 2962	. . . . . . . . 9 ⊢ (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
44	43	anbi2i 635	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45		annim 409	. . . . . . . 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 7877	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
48		nnon 7877	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
49		onsssuc 6460	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦))
50		onsuc 7818	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
51		onelpss 6408	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
52	50, 51	sylan2 605	. . . . . . . . . 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 609	. . . . . . . 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 418	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
57	56	a1ddd 81	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
58	57	a2d 30	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ 𝑦 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
59	58	com23 87	. . . . . . . 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	biimtrrid 246	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
62	42, 61	pm2.61d 181	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃)))
63	62	ex 418	. . . 4 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
64	63	a2d 30	. . 3 ⊢ (𝑦 ∈ ω → ((𝐵 ∈ ω → (𝐵 ⊆ 𝑦 → 𝜒)) → (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦 → 𝜃))))
65	15, 19, 23, 27, 29, 64	finds 7902	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵 ⊆ 𝐴 → 𝜏)))
66	65	imp31 423	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → 𝜏)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2961   ⊆ wss 3908  ∅c0 4289  Oncon0 6367  suc csuc 6369  ωcom 7871

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-om 7872

This theorem is used by:  nnaordi  8613  inf3lem5  9611  ackbij2lem4  10243  sornom  10279  fin23lem15  10336  fin23lem36  10350  isf32lem1  10355  isf32lem2  10356  wunex2  10741  indpi  10910  satfsschain  35877