Theorem tfindsg2 2403

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 suc B instead of zero.

Hypotheses

Ref	Expression
tfindsg2.1	⊢ (x = suc B → (φ ↔ ψ))
tfindsg2.2	⊢ (x = y → (φ ↔ χ))
tfindsg2.3	⊢ (x = suc y → (φ ↔ θ))
tfindsg2.4	⊢ (x = A → (φ ↔ τ))
tfindsg2.5	⊢ (B ∈ On → ψ)
tfindsg2.6	⊢ ((y ∈ On ∧ B ∈ y) → (χ → θ))
tfindsg2.7	⊢ ((Lim x ∧ B ∈ x) → (∀y ∈ x (B ∈ y → χ) → φ))

Assertion

Ref	Expression
tfindsg2	⊢ ((A ∈ 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 tfindsg2

Step	Hyp	Ref	Expression
1		onelon 2223	. . 3 ⊢ ((A ∈ On ∧ B ∈ A) → B ∈ On)
2		sucelon 2319	. . 3 ⊢ (B ∈ On ↔ suc B ∈ On)
3	1, 2	sylib 173	. 2 ⊢ ((A ∈ On ∧ B ∈ A) → suc B ∈ On)
4		eloni 2209	. . . 4 ⊢ (A ∈ On → Ord A)
5		ordsucss 2320	. . . 4 ⊢ (Ord A → (B ∈ A → suc B ⊆ A))
6	4, 5	syl 12	. . 3 ⊢ (A ∈ On → (B ∈ A → suc B ⊆ A))
7	6	imp 277	. 2 ⊢ ((A ∈ On ∧ B ∈ A) → suc B ⊆ A)
8		tfindsg2.1	. . . . . 6 ⊢ (x = suc B → (φ ↔ ψ))
9		tfindsg2.2	. . . . . 6 ⊢ (x = y → (φ ↔ χ))
10		tfindsg2.3	. . . . . 6 ⊢ (x = suc y → (φ ↔ θ))
11		tfindsg2.4	. . . . . 6 ⊢ (x = A → (φ ↔ τ))
12		tfindsg2.5	. . . . . . 7 ⊢ (B ∈ On → ψ)
13	2, 12	sylbir 176	. . . . . 6 ⊢ (suc B ∈ On → ψ)
14		ordelsuc 2322	. . . . . . . . . . 11 ⊢ ((B ∈ On ∧ Ord y) → (B ∈ y ↔ suc B ⊆ y))
15		eloni 2209	. . . . . . . . . . 11 ⊢ (y ∈ On → Ord y)
16	14, 15	sylan2 346	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ y ∈ On) → (B ∈ y ↔ suc B ⊆ y))
17	16	ancoms 334	. . . . . . . . 9 ⊢ ((y ∈ On ∧ B ∈ On) → (B ∈ y ↔ suc B ⊆ y))
18		tfindsg2.6	. . . . . . . . . . 11 ⊢ ((y ∈ On ∧ B ∈ y) → (χ → θ))
19	18	exp 291	. . . . . . . . . 10 ⊢ (y ∈ On → (B ∈ y → (χ → θ)))
20	19	adantr 306	. . . . . . . . 9 ⊢ ((y ∈ On ∧ B ∈ On) → (B ∈ y → (χ → θ)))
21	17, 20	sylbird 180	. . . . . . . 8 ⊢ ((y ∈ On ∧ B ∈ On) → (suc B ⊆ y → (χ → θ)))
22	21, 2	sylan2br 348	. . . . . . 7 ⊢ ((y ∈ On ∧ suc B ∈ On) → (suc B ⊆ y → (χ → θ)))
23	22	imp 277	. . . . . 6 ⊢ (((y ∈ On ∧ suc B ∈ On) ∧ suc B ⊆ y) → (χ → θ))
24		tfindsg2.7	. . . . . . . . . . 11 ⊢ ((Lim x ∧ B ∈ x) → (∀y ∈ x (B ∈ y → χ) → φ))
25	24	exp 291	. . . . . . . . . 10 ⊢ (Lim x → (B ∈ x → (∀y ∈ x (B ∈ y → χ) → φ)))
26	25	adantr 306	. . . . . . . . 9 ⊢ ((Lim x ∧ B ∈ On) → (B ∈ x → (∀y ∈ x (B ∈ y → χ) → φ)))
27		ordelsuc 2322	. . . . . . . . . . . . 13 ⊢ ((B ∈ On ∧ Ord x) → (B ∈ x ↔ suc B ⊆ x))
28		eloni 2209	. . . . . . . . . . . . 13 ⊢ (x ∈ On → Ord x)
29	27, 28	sylan2 346	. . . . . . . . . . . 12 ⊢ ((B ∈ On ∧ x ∈ On) → (B ∈ x ↔ suc B ⊆ x))
30		onelon 2223	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∈ On ∧ y ∈ x) → y ∈ On)
31	30, 15	syl 12	. . . . . . . . . . . . . . . . 17 ⊢ ((x ∈ On ∧ y ∈ x) → Ord y)
32	14, 31	sylan2 346	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ On ∧ (x ∈ On ∧ y ∈ x)) → (B ∈ y ↔ suc B ⊆ y))
33	32	anassrs 338	. . . . . . . . . . . . . . 15 ⊢ (((B ∈ On ∧ x ∈ On) ∧ y ∈ x) → (B ∈ y ↔ suc B ⊆ y))
34	33	imbi1d 465	. . . . . . . . . . . . . 14 ⊢ (((B ∈ On ∧ x ∈ On) ∧ y ∈ x) → ((B ∈ y → χ) ↔ (suc B ⊆ y → χ)))
35	34	biraldva 1215	. . . . . . . . . . . . 13 ⊢ ((B ∈ On ∧ x ∈ On) → (∀y ∈ x (B ∈ y → χ) ↔ ∀y ∈ x (suc B ⊆ y → χ)))
36	35	imbi1d 465	. . . . . . . . . . . 12 ⊢ ((B ∈ On ∧ x ∈ On) → ((∀y ∈ x (B ∈ y → χ) → φ) ↔ (∀y ∈ x (suc B ⊆ y → χ) → φ)))
37	29, 36	imbi12d 474	. . . . . . . . . . 11 ⊢ ((B ∈ On ∧ x ∈ On) → ((B ∈ x → (∀y ∈ x (B ∈ y → χ) → φ)) ↔ (suc B ⊆ x → (∀y ∈ x (suc B ⊆ y → χ) → φ))))
38		visset 1350	. . . . . . . . . . . 12 ⊢ x ∈ V
39		limelon 2286	. . . . . . . . . . . 12 ⊢ ((x ∈ V ∧ Lim x) → x ∈ On)
40	38, 39	mpan 518	. . . . . . . . . . 11 ⊢ (Lim x → x ∈ On)
41	37, 40	sylan2 346	. . . . . . . . . 10 ⊢ ((B ∈ On ∧ Lim x) → ((B ∈ x → (∀y ∈ x (B ∈ y → χ) → φ)) ↔ (suc B ⊆ x → (∀y ∈ x (suc B ⊆ y → χ) → φ))))
42	41	ancoms 334	. . . . . . . . 9 ⊢ ((Lim x ∧ B ∈ On) → ((B ∈ x → (∀y ∈ x (B ∈ y → χ) → φ)) ↔ (suc B ⊆ x → (∀y ∈ x (suc B ⊆ y → χ) → φ))))
43	26, 42	mpbid 170	. . . . . . . 8 ⊢ ((Lim x ∧ B ∈ On) → (suc B ⊆ x → (∀y ∈ x (suc B ⊆ y → χ) → φ)))
44	43, 2	sylan2br 348	. . . . . . 7 ⊢ ((Lim x ∧ suc B ∈ On) → (suc B ⊆ x → (∀y ∈ x (suc B ⊆ y → χ) → φ)))
45	44	imp 277	. . . . . 6 ⊢ (((Lim x ∧ suc B ∈ On) ∧ suc B ⊆ x) → (∀y ∈ x (suc B ⊆ y → χ) → φ))
46	8, 9, 10, 11, 13, 23, 45	tfindsg 2402	. . . . 5 ⊢ (((A ∈ On ∧ suc B ∈ On) ∧ suc B ⊆ A) → τ)
47	46	exp31 293	. . . 4 ⊢ (A ∈ On → (suc B ∈ On → (suc B ⊆ A → τ)))
48	47	imp3a 279	. . 3 ⊢ (A ∈ On → ((suc B ∈ On ∧ suc B ⊆ A) → τ))
49	48	adantr 306	. 2 ⊢ ((A ∈ On ∧ B ∈ A) → ((suc B ∈ On ∧ suc B ⊆ A) → τ))
50	3, 7, 49	mp2and 526	1 ⊢ ((A ∈ On ∧ B ∈ A) → τ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

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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