Theorem tfindsg 7794

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 3962	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
2	1	adantl 481	. . . . . 6 ⊢ ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ ∅))
3		eqeq2 2741	. . . . . . . 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 4353	. . . . . . . . 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 811	. . . 4 ⊢ (𝑥 = ∅ → ((𝐵 ⊆ 𝑥 → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
15	14	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ On → (𝐵 ⊆ 𝑥 → 𝜑)) ↔ (𝐵 ∈ On → (𝐵 ⊆ ∅ → 𝜓))))
16		sseq2 3962	. . . . 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 3962	. . . . 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 3962	. . . . 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 3440	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
31	30	sucex 7742	. . . . . . . . . . . . 13 ⊢ suc 𝑦 ∈ V
32	31	eqvinc 3604	. . . . . . . . . . . 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 1930	. . . . . . . . . . . 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 2737	. . . . . . . . . 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 2926	. . . . . . . . 9 ⊢ (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
44	43	anbi2i 623	. . . . . . . 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 6399	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ∈ suc 𝑦))
48		onsuc 7746	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49		onelpss 6347	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦 ∧ 𝐵 ≠ suc 𝑦)))
50	48, 49	sylan2 593	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑦 ∈ 𝑥 (𝐵 ∈ On → (𝐵 ⊆ 𝑦 → 𝜒)) → ∀𝑦 ∈ 𝑥 (𝐵 ⊆ 𝑦 → 𝜒)))
65	64	ad2antlr 727	. . . . . . 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 7793	. 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3903  ∅c0 4284  Oncon0 6307  Lim wlim 6308  suc csuc 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313

This theorem is referenced by:  tfindsg2  7795  oaordi  8464  infensuc  9072  r1ordg  9674