Theorem tfindsg 2402

Description: Transfinite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal B instead of zero. Remark of [TakeutiZaring] p. 57.

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 group(s):   x,A   x,y,B   ψ,x   χ,x   θ,x   τ,x   φ,y

Proof of Theorem tfindsg

Step	Hyp	Ref	Expression
1		sseq2 1522	. . . . . . 7 ⊢ (x = ∅ → (B ⊆ x ↔ B ⊆ ∅))
2	1	adantl 305	. . . . . 6 ⊢ ((B = ∅ ∧ x = ∅) → (B ⊆ x ↔ B ⊆ ∅))
3		cleq2 1110	. . . . . . . 8 ⊢ (B = ∅ → (x = B ↔ x = ∅))
4		tfindsg.1	. . . . . . . 8 ⊢ (x = B → (φ ↔ ψ))
5	3, 4	syl6bir 188	. . . . . . 7 ⊢ (B = ∅ → (x = ∅ → (φ ↔ ψ)))
6	5	imp 277	. . . . . 6 ⊢ ((B = ∅ ∧ x = ∅) → (φ ↔ ψ))
7	2, 6	imbi12d 474	. . . . 5 ⊢ ((B = ∅ ∧ x = ∅) → ((B ⊆ x → φ) ↔ (B ⊆ ∅ → ψ)))
8	1	imbi1d 465	. . . . . 6 ⊢ (x = ∅ → ((B ⊆ x → φ) ↔ (B ⊆ ∅ → φ)))
9		ss0 1727	. . . . . . . . 9 ⊢ (B ⊆ ∅ → B = ∅)
10	9	con3i 90	. . . . . . . 8 ⊢ (¬ B = ∅ → ¬ B ⊆ ∅)
11	10	pm2.21d 74	. . . . . . 7 ⊢ (¬ B = ∅ → (B ⊆ ∅ → (φ ↔ ψ)))
12	11	pm5.74d 444	. . . . . 6 ⊢ (¬ B = ∅ → ((B ⊆ ∅ → φ) ↔ (B ⊆ ∅ → ψ)))
13	8, 12	sylan9bbr 419	. . . . 5 ⊢ ((¬ B = ∅ ∧ x = ∅) → ((B ⊆ x → φ) ↔ (B ⊆ ∅ → ψ)))
14	7, 13	pm2.61an1 364	. . . 4 ⊢ (x = ∅ → ((B ⊆ x → φ) ↔ (B ⊆ ∅ → ψ)))
15	14	imbi2d 464	. . 3 ⊢ (x = ∅ → ((B ∈ On → (B ⊆ x → φ)) ↔ (B ∈ On → (B ⊆ ∅ → ψ))))
16		sseq2 1522	. . . . 5 ⊢ (x = y → (B ⊆ x ↔ B ⊆ y))
17		tfindsg.2	. . . . 5 ⊢ (x = y → (φ ↔ χ))
18	16, 17	imbi12d 474	. . . 4 ⊢ (x = y → ((B ⊆ x → φ) ↔ (B ⊆ y → χ)))
19	18	imbi2d 464	. . 3 ⊢ (x = y → ((B ∈ On → (B ⊆ x → φ)) ↔ (B ∈ On → (B ⊆ y → χ))))
20		sseq2 1522	. . . . 5 ⊢ (x = suc y → (B ⊆ x ↔ B ⊆ suc y))
21		tfindsg.3	. . . . 5 ⊢ (x = suc y → (φ ↔ θ))
22	20, 21	imbi12d 474	. . . 4 ⊢ (x = suc y → ((B ⊆ x → φ) ↔ (B ⊆ suc y → θ)))
23	22	imbi2d 464	. . 3 ⊢ (x = suc y → ((B ∈ On → (B ⊆ x → φ)) ↔ (B ∈ On → (B ⊆ suc y → θ))))
24		sseq2 1522	. . . . 5 ⊢ (x = A → (B ⊆ x ↔ B ⊆ A))
25		tfindsg.4	. . . . 5 ⊢ (x = A → (φ ↔ τ))
26	24, 25	imbi12d 474	. . . 4 ⊢ (x = A → ((B ⊆ x → φ) ↔ (B ⊆ A → τ)))
27	26	imbi2d 464	. . 3 ⊢ (x = A → ((B ∈ On → (B ⊆ x → φ)) ↔ (B ∈ On → (B ⊆ A → τ))))
28		tfindsg.5	. . . 4 ⊢ (B ∈ On → ψ)
29	28	a1d 14	. . 3 ⊢ (B ∈ On → (B ⊆ ∅ → ψ))
30		visset 1350	. . . . . . . . . . . . . 14 ⊢ y ∈ V
31	30	sucex 2303	. . . . . . . . . . . . 13 ⊢ suc y ∈ V
32	31	eqvinc 1407	. . . . . . . . . . . 12 ⊢ (suc y = B ↔ ∃x(x = suc y ∧ x = B))
33	4, 28	syl5bir 184	. . . . . . . . . . . . . 14 ⊢ (x = B → (B ∈ On → φ))
34	21	biimpd 135	. . . . . . . . . . . . . 14 ⊢ (x = suc y → (φ → θ))
35	33, 34	sylan9r 360	. . . . . . . . . . . . 13 ⊢ ((x = suc y ∧ x = B) → (B ∈ On → θ))
36	35	19.23aiv 952	. . . . . . . . . . . 12 ⊢ (∃x(x = suc y ∧ x = B) → (B ∈ On → θ))
37	32, 36	sylbi 174	. . . . . . . . . . 11 ⊢ (suc y = B → (B ∈ On → θ))
38	37	cleqcoms 1104	. . . . . . . . . 10 ⊢ (B = suc y → (B ∈ On → θ))
39	38	syl3 18	. . . . . . . . 9 ⊢ ((B ⊆ suc y → B = suc y) → (B ⊆ suc y → (B ∈ On → θ)))
40	39	a1d 14	. . . . . . . 8 ⊢ ((B ⊆ suc y → B = suc y) → ((B ⊆ y → χ) → (B ⊆ suc y → (B ∈ On → θ))))
41	40	com4r 41	. . . . . . 7 ⊢ (B ∈ On → ((B ⊆ suc y → B = suc y) → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
42	41	adantl 305	. . . . . 6 ⊢ ((y ∈ On ∧ B ∈ On) → ((B ⊆ suc y → B = suc y) → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
43		onsssuc 2311	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ y ∈ On) → (B ⊆ y ↔ B ∈ suc y))
44		onelpsst 2253	. . . . . . . . . . 11 ⊢ ((B ∈ On ∧ suc y ∈ On) → (B ∈ suc y ↔ (B ⊆ suc y ∧ ¬ B = suc y)))
45		suceloni 2314	. . . . . . . . . . 11 ⊢ (y ∈ On → suc y ∈ On)
46	44, 45	sylan2 346	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ y ∈ On) → (B ∈ suc y ↔ (B ⊆ suc y ∧ ¬ B = suc y)))
47	43, 46	bitrd 406	. . . . . . . . 9 ⊢ ((B ∈ On ∧ y ∈ On) → (B ⊆ y ↔ (B ⊆ suc y ∧ ¬ B = suc y)))
48	47	ancoms 334	. . . . . . . 8 ⊢ ((y ∈ On ∧ B ∈ On) → (B ⊆ y ↔ (B ⊆ suc y ∧ ¬ B = suc y)))
49		tfindsg.6	. . . . . . . . . . . 12 ⊢ (((y ∈ On ∧ B ∈ On) ∧ B ⊆ y) → (χ → θ))
50	49	exp 291	. . . . . . . . . . 11 ⊢ ((y ∈ On ∧ B ∈ On) → (B ⊆ y → (χ → θ)))
51		ax-1 3	. . . . . . . . . . 11 ⊢ (θ → (B ⊆ suc y → θ))
52	50, 51	syl8 25	. . . . . . . . . 10 ⊢ ((y ∈ On ∧ B ∈ On) → (B ⊆ y → (χ → (B ⊆ suc y → θ))))
53	52	a2d 15	. . . . . . . . 9 ⊢ ((y ∈ On ∧ B ∈ On) → ((B ⊆ y → χ) → (B ⊆ y → (B ⊆ suc y → θ))))
54	53	com23 32	. . . . . . . 8 ⊢ ((y ∈ On ∧ B ∈ On) → (B ⊆ y → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
55	48, 54	sylbird 180	. . . . . . 7 ⊢ ((y ∈ On ∧ B ∈ On) → ((B ⊆ suc y ∧ ¬ B = suc y) → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
56		annim 206	. . . . . . 7 ⊢ ((B ⊆ suc y ∧ ¬ B = suc y) ↔ ¬ (B ⊆ suc y → B = suc y))
57	55, 56	syl5ibr 182	. . . . . 6 ⊢ ((y ∈ On ∧ B ∈ On) → (¬ (B ⊆ suc y → B = suc y) → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
58	42, 57	pm2.61d 112	. . . . 5 ⊢ ((y ∈ On ∧ B ∈ On) → ((B ⊆ y → χ) → (B ⊆ suc y → θ)))
59	58	exp 291	. . . 4 ⊢ (y ∈ On → (B ∈ On → ((B ⊆ y → χ) → (B ⊆ suc y → θ))))
60	59	a2d 15	. . 3 ⊢ (y ∈ On → ((B ∈ On → (B ⊆ y → χ)) → (B ∈ On → (B ⊆ suc y → θ))))
61		pm2.27 30	. . . . . . . . 9 ⊢ (B ∈ On → ((B ∈ On → (B ⊆ y → χ)) → (B ⊆ y → χ)))
62	61	r19.20sdv 1257	. . . . . . . 8 ⊢ (B ∈ On → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → ∀y ∈ x (B ⊆ y → χ)))
63	62	ad2antlr 321	. . . . . . 7 ⊢ (((Lim x ∧ B ∈ On) ∧ B ⊆ x) → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → ∀y ∈ x (B ⊆ y → χ)))
64		tfindsg.7	. . . . . . 7 ⊢ (((Lim x ∧ B ∈ On) ∧ B ⊆ x) → (∀y ∈ x (B ⊆ y → χ) → φ))
65	63, 64	syld 27	. . . . . 6 ⊢ (((Lim x ∧ B ∈ On) ∧ B ⊆ x) → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → φ))
66	65	exp31 293	. . . . 5 ⊢ (Lim x → (B ∈ On → (B ⊆ x → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → φ))))
67	66	com3l 34	. . . 4 ⊢ (B ∈ On → (B ⊆ x → (Lim x → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → φ))))
68	67	com4t 40	. . 3 ⊢ (Lim x → (∀y ∈ x (B ∈ On → (B ⊆ y → χ)) → (B ∈ On → (B ⊆ x → φ))))
69	15, 19, 23, 27, 29, 60, 68	tfinds 2401	. 2 ⊢ (A ∈ On → (B ∈ On → (B ⊆ A → τ)))
70	69	imp31 280	1 ⊢ (((A ∈ On ∧ B ∈ On) ∧ B ⊆ A) → τ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∅c0 1707  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  tfindsg2 2403  oaordi 3148  infensuc 3484  r1ord 3499

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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