Theorem tfindsg 7854

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 𝐵 instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.)

Hypotheses

Ref	Expression
tfindsg.1	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
tfindsg.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
tfindsg.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
tfindsg.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
tfindsg.5	⊢ (𝐵 ∈ On → 𝜓)
tfindsg.6	⊢ (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))
tfindsg.7	⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))

Assertion

Ref	Expression
tfindsg	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 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 tfindsg

Step	Hyp	Ref	Expression
1		sseq2 4008	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
2	1	adantl 481	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
3		eqeq2 2743	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝑥 = 𝐵 ↔ 𝑥 = ∅))
4		tfindsg.1	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
5	3, 4	syl6bir 254	. . . . . . 7 ⊢ (𝐵 = ∅ → (𝑥 = ∅ → (𝜑 ↔ 𝜓)))
6	5	imp 406	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑 ↔ 𝜓))
7	2, 6	imbi12d 344	. . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
8	1	imbi1d 341	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9		ss0 4398	. . . . . . . . 9 ⊢ (𝐵 ⊆ ∅ → 𝐵 = ∅)
10	9	con3i 154	. . . . . . . 8 ⊢ (¬ 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
11	10	pm2.21d 121	. . . . . . 7 ⊢ (¬ 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑 ↔ 𝜓)))
12	11	pm5.74d 273	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
13	8, 12	sylan9bbr 510	. . . . 5 ⊢ ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
14	7, 13	pm2.61ian 809	. . . 4 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
15	14	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16		sseq2 4008	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦))
17		tfindsg.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
18	16, 17	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝑦 → 𝜒)))
19	18	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒))))
20		sseq2 4008	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ suc 𝑦))
21		tfindsg.3	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
22	20, 21	imbi12d 344	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ suc 𝑦 → 𝜃)))
23	22	imbi2d 340	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦 → 𝜃))))
24		sseq2 4008	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝐴))
25		tfindsg.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
26	24, 25	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ 𝐴 → 𝜏)))
27	26	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ 𝐴 → 𝜏))))
28		tfindsg.5	. . . 4 ⊢ (𝐵 ∈ On → 𝜓)
29	28	a1d 25	. . 3 ⊢ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))
30		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
31	30	sucex 7798	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ∈ V
32	31	eqvinc 3637	. . . . . . . . . . . 12 ⊢ (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵))
33	28, 4	imbitrrid 245	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
34	21	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → (𝜑 → 𝜃))
35	33, 34	sylan9r 508	. . . . . . . . . . . . 13 ⊢ ((𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
36	35	exlimiv 1932	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵) → (𝐵 ∈ On → 𝜃))
37	32, 36	sylbi 216	. . . . . . . . . . 11 ⊢ (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
38	37	eqcoms 2739	. . . . . . . . . 10 ⊢ (𝐵 = suc 𝑦 → (𝐵 ∈ On → 𝜃))
39	38	imim2i 16	. . . . . . . . 9 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ On → 𝜃)))
40	39	a1d 25	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ On → 𝜃))))
41	40	com4r 94	. . . . . . 7 ⊢ (𝐵 ∈ On → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
42	41	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
43		df-ne 2940	. . . . . . . . 9 ⊢ (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
44	43	anbi2i 622	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45		annim 403	. . . . . . . 8 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦))
46	44, 45	bitri 275	. . . . . . 7 ⊢ ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦))
47		onsssuc 6454	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦))
48		onsuc 7803	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49		onelpss 6404	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
50	48, 49	sylan2 592	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
51	47, 50	bitrd 279	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
52	51	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
53		tfindsg.6	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑦) → (𝜒 → 𝜃))
54	53	ex 412	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝑦 → (𝜒 → 𝜃)))
55	54	a1ddd 80	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
56	55	a2d 29	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ 𝑦 → (𝐵 ⊆ suc 𝑦 → 𝜃))))
57	56	com23 86	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝑦 → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
58	52, 57	sylbird 260	. . . . . . 7 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
59	46, 58	biimtrrid 242	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐵 ⊆ suc 𝑦 → 𝐵 = suc 𝑦) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
60	42, 59	pm2.61d 179	. . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃)))
61	60	ex 412	. . . 4 ⊢ (𝑦 ∈ On → (𝐵 ∈ On → ((𝐵 ⊆ 𝑦 → 𝜒) → (𝐵 ⊆ suc 𝑦 → 𝜃))))
62	61	a2d 29	. . 3 ⊢ (𝑦 ∈ On → ((𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → (𝐵 ∈ On → (𝐵 ⊆ suc 𝑦 → 𝜃))))
63		pm2.27 42	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ((𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → (𝐵 ⊆ 𝑦 → 𝜒)))
64	63	ralimdv 3168	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → ∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒)))
65	64	ad2antlr 724	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → ∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒)))
66		tfindsg.7	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒) → 𝜑))
67	65, 66	syld 47	. . . . . 6 ⊢ (((Lim 𝑥 ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → 𝜑))
68	67	exp31 419	. . . . 5 ⊢ (Lim 𝑥 → (𝐵 ∈ On → (𝐵 ⊆ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → 𝜑))))
69	68	com3l 89	. . . 4 ⊢ (𝐵 ∈ On → (𝐵 ⊆ 𝑥 → (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → 𝜑))))
70	69	com4t 93	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → (𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑))))
71	15, 19, 23, 27, 29, 62, 70	tfinds 7853	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐵 ⊆ 𝐴 → 𝜏)))
72	71	imp31 417	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 ⊆ 𝐴) → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060   ⊆ wss 3948  ∅c0 4322  Oncon0 6364  Lim wlim 6365  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370

This theorem is referenced by:  tfindsg2  7855  oaordi  8552  infensuc  9161  r1ordg  9779