Theorem tfindsg 7806

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 3949	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
2	1	adantl 481	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
3		eqeq2 2749	. . . . . . . 8 ⊢ (𝐵 = ∅ → (𝑥 = 𝐵 ↔ 𝑥 = ∅))
4		tfindsg.1	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
5	3, 4	biimtrrdi 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 4343	. . . . . . . . 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 812	. . . 4 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
15	14	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16		sseq2 3949	. . . . 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 3949	. . . . 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 3949	. . . . 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 3434	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
31	30	sucex 7754	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ∈ V
32	31	eqvinc 3592	. . . . . . . . . . . 12 ⊢ (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦 ∧ 𝑥 = 𝐵))
33	28, 4	imbitrrid 246	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝐵 ∈ On → 𝜑))
34	21	biimpd 229	. . . . . . . . . . . . . 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 217	. . . . . . . . . . 11 ⊢ (suc 𝑦 = 𝐵 → (𝐵 ∈ On → 𝜃))
38	37	eqcoms 2745	. . . . . . . . . 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 2934	. . . . . . . . 9 ⊢ (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
44	43	anbi2i 624	. . . . . . . 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 6410	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦))
48		onsuc 7758	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49		onelpss 6358	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
50	48, 49	sylan2 594	. . . . . . . . . 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 243	. . . . . 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 3152	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → ∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒)))
65	64	ad2antlr 728	. . . . . . 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 7805	. 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 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3890  ∅c0 4274  Oncon0 6318  Lim wlim 6319  suc csuc 6320

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324

This theorem is referenced by:  tfindsg2  7807  oaordi  8475  infensuc  9087  r1ordg  9696