Theorem tfinds 2401

Description: Principle of 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. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

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

Assertion

Ref	Expression
tfinds	⊢ (A ∈ On → τ)

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

Proof of Theorem tfinds

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 ⊢ (x = y → (φ ↔ χ))
2		tfinds.4	. 2 ⊢ (x = A → (φ ↔ τ))
3		eloni 2209	. . . . . 6 ⊢ (x ∈ On → Ord x)
4		df-lim 2204	. . . . . . . . . . . 12 ⊢ (Lim x ↔ (Ord x ∧ ¬ x = ∅ ∧ x = ∪x))
5	4	biimpr 134	. . . . . . . . . . 11 ⊢ ((Ord x ∧ ¬ x = ∅ ∧ x = ∪x) → Lim x)
6	5	3exp 611	. . . . . . . . . 10 ⊢ (Ord x → (¬ x = ∅ → (x = ∪x → Lim x)))
7		con3 86	. . . . . . . . . 10 ⊢ ((x = ∪x → Lim x) → (¬ Lim x → ¬ x = ∪x))
8	6, 7	syl6 23	. . . . . . . . 9 ⊢ (Ord x → (¬ x = ∅ → (¬ Lim x → ¬ x = ∪x)))
9	8	com23 32	. . . . . . . 8 ⊢ (Ord x → (¬ Lim x → (¬ x = ∅ → ¬ x = ∪x)))
10		orduninsuc 2365	. . . . . . . . . 10 ⊢ (Ord x → (x = ∪x ↔ ¬ ∃y ∈ On x = suc y))
11	10	biimprd 136	. . . . . . . . 9 ⊢ (Ord x → (¬ ∃y ∈ On x = suc y → x = ∪x))
12	11	con1d 85	. . . . . . . 8 ⊢ (Ord x → (¬ x = ∪x → ∃y ∈ On x = suc y))
13	9, 12	syl6d 54	. . . . . . 7 ⊢ (Ord x → (¬ Lim x → (¬ x = ∅ → ∃y ∈ On x = suc y)))
14		df-or 197	. . . . . . 7 ⊢ ((x = ∅ ∨ ∃y ∈ On x = suc y) ↔ (¬ x = ∅ → ∃y ∈ On x = suc y))
15	13, 14	syl6ibr 186	. . . . . 6 ⊢ (Ord x → (¬ Lim x → (x = ∅ ∨ ∃y ∈ On x = suc y)))
16	3, 15	syl 12	. . . . 5 ⊢ (x ∈ On → (¬ Lim x → (x = ∅ ∨ ∃y ∈ On x = suc y)))
17		tfinds.5	. . . . . . . . 9 ⊢ ψ
18		tfinds.1	. . . . . . . . 9 ⊢ (x = ∅ → (φ ↔ ψ))
19	17, 18	mpbiri 169	. . . . . . . 8 ⊢ (x = ∅ → φ)
20	19	a1d 14	. . . . . . 7 ⊢ (x = ∅ → (∀y ∈ x χ → φ))
21	20	a1d 14	. . . . . 6 ⊢ (x = ∅ → (x ∈ On → (∀y ∈ x χ → φ)))
22		ax-17 925	. . . . . . . 8 ⊢ (x ∈ On → ∀y x ∈ On)
23		hbra1 1237	. . . . . . . . 9 ⊢ (∀y ∈ x χ → ∀y∀y ∈ x χ)
24		ax-17 925	. . . . . . . . 9 ⊢ (φ → ∀yφ)
25	23, 24	hbim 702	. . . . . . . 8 ⊢ ((∀y ∈ x χ → φ) → ∀y(∀y ∈ x χ → φ))
26	22, 25	hbim 702	. . . . . . 7 ⊢ ((x ∈ On → (∀y ∈ x χ → φ)) → ∀y(x ∈ On → (∀y ∈ x χ → φ)))
27		raleq 1324	. . . . . . . . . . . . 13 ⊢ (x = suc y → (∀z ∈ x [z / x]φ ↔ ∀z ∈ suc y[z / x]φ))
28		sbequ 877	. . . . . . . . . . . . . . 15 ⊢ (y = z → ([y / x]φ ↔ [z / x]φ))
29		ax-17 925	. . . . . . . . . . . . . . . 16 ⊢ (χ → ∀xχ)
30	29, 1	sbie 904	. . . . . . . . . . . . . . 15 ⊢ ([y / x]φ ↔ χ)
31	28, 30	syl5bbr 412	. . . . . . . . . . . . . 14 ⊢ (y = z → (χ ↔ [z / x]φ))
32	31	cbvralv 1333	. . . . . . . . . . . . 13 ⊢ (∀y ∈ x χ ↔ ∀z ∈ x [z / x]φ)
33		ax-17 925	. . . . . . . . . . . . . 14 ⊢ (φ → ∀zφ)
34		hbs1 986	. . . . . . . . . . . . . 14 ⊢ ([z / x]φ → ∀x[z / x]φ)
35		sbequ12 865	. . . . . . . . . . . . . 14 ⊢ (x = z → (φ ↔ [z / x]φ))
36	33, 34, 35	cbvral 1331	. . . . . . . . . . . . 13 ⊢ (∀x ∈ suc yφ ↔ ∀z ∈ suc y[z / x]φ)
37	27, 32, 36	3bitr4g 428	. . . . . . . . . . . 12 ⊢ (x = suc y → (∀y ∈ x χ ↔ ∀x ∈ suc yφ))
38	37	biimpd 135	. . . . . . . . . . 11 ⊢ (x = suc y → (∀y ∈ x χ → ∀x ∈ suc yφ))
39		tfinds.6	. . . . . . . . . . . 12 ⊢ (y ∈ On → (χ → θ))
40		visset 1350	. . . . . . . . . . . . . 14 ⊢ y ∈ V
41	40	sucid 2304	. . . . . . . . . . . . 13 ⊢ y ∈ suc y
42	1	rcla4v 1402	. . . . . . . . . . . . 13 ⊢ (∀x ∈ suc yφ → (y ∈ suc y → χ))
43	41, 42	mpi 44	. . . . . . . . . . . 12 ⊢ (∀x ∈ suc yφ → χ)
44	39, 43	syl5 22	. . . . . . . . . . 11 ⊢ (y ∈ On → (∀x ∈ suc yφ → θ))
45	38, 44	sylan9r 360	. . . . . . . . . 10 ⊢ ((y ∈ On ∧ x = suc y) → (∀y ∈ x χ → θ))
46		tfinds.3	. . . . . . . . . . 11 ⊢ (x = suc y → (φ ↔ θ))
47	46	adantl 305	. . . . . . . . . 10 ⊢ ((y ∈ On ∧ x = suc y) → (φ ↔ θ))
48	45, 47	sylibrd 179	. . . . . . . . 9 ⊢ ((y ∈ On ∧ x = suc y) → (∀y ∈ x χ → φ))
49	48	a1d 14	. . . . . . . 8 ⊢ ((y ∈ On ∧ x = suc y) → (x ∈ On → (∀y ∈ x χ → φ)))
50	49	exp 291	. . . . . . 7 ⊢ (y ∈ On → (x = suc y → (x ∈ On → (∀y ∈ x χ → φ))))
51	26, 50	r19.23ai 1283	. . . . . 6 ⊢ (∃y ∈ On x = suc y → (x ∈ On → (∀y ∈ x χ → φ)))
52	21, 51	jaoi 275	. . . . 5 ⊢ ((x = ∅ ∨ ∃y ∈ On x = suc y) → (x ∈ On → (∀y ∈ x χ → φ)))
53	16, 52	syl6 23	. . . 4 ⊢ (x ∈ On → (¬ Lim x → (x ∈ On → (∀y ∈ x χ → φ))))
54	53	pm2.43a 60	. . 3 ⊢ (x ∈ On → (¬ Lim x → (∀y ∈ x χ → φ)))
55		tfinds.7	. . 3 ⊢ (Lim x → (∀y ∈ x χ → φ))
56	54, 55	pm2.61d2 111	. 2 ⊢ (x ∈ On → (∀y ∈ x χ → φ))
57	1, 2, 56	tfis3 2248	1 ⊢ (A ∈ On → τ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = weq 797  [wsb 852   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  tfindsg 2402  tfindes 2404  tfinds3 2406  oa0r 3141  om0r 3142  om1r 3145  oe1m 3147  r1tr 3498  alephon 3671  alephcard 3673  alephordi 3679

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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