Theorem tfinds 7842

Description: Principle of 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. Theorem Schema 4 of [Suppes] p. 197. Theorem 1.19 of [Schloeder] p. 3. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

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 groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfinds

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
2		tfinds.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
3		dflim3 7829	. . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
4	3	notbii 322	. . . 4 ⊢ (¬ Lim 𝑥 ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
5		iman 405	. . . . 5 ⊢ ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6		eloni 6358	. . . . . . 7 ⊢ (𝑥 ∈ On → Ord 𝑥)
7		pm2.27 42	. . . . . . 7 ⊢ (Ord 𝑥 → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
8	6, 7	syl 17	. . . . . 6 ⊢ (𝑥 ∈ On → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
9		tfinds.5	. . . . . . . . 9 ⊢ 𝜓
10		tfinds.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
11	9, 10	mpbiri 260	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
12	11	a1d 25	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
13		nfra1 3288	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑥 𝜒
14		nfv 1936	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
15	13, 14	nfim 1918	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝑥 𝜒 → 𝜑)
16		vex 3460	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	16	sucid 6432	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ suc 𝑦
18	1	rspcv 3579	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ suc 𝑦 → (∀𝑥 ∈ suc 𝑦𝜑 → 𝜒))
19	17, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ suc 𝑦𝜑 → 𝜒)
20		tfinds.6	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝜒 → 𝜃))
21	19, 20	syl5 34	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (∀𝑥 ∈ suc 𝑦𝜑 → 𝜃))
22		raleq 3319	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23		nfv 1936	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝜒
24	23, 1	sbiev 2348	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
25		sbequ 2118	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
26	24, 25	bitr3id 287	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
27	26	cbvralvw 3242	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
28		cbvralsvw 3315	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ suc 𝑦𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑)
29	22, 27, 28	3bitr4g 316	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑥 ∈ suc 𝑦𝜑))
30	29	imbi1d 343	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((∀𝑦 ∈ 𝑥 𝜒 → 𝜃) ↔ (∀𝑥 ∈ suc 𝑦𝜑 → 𝜃)))
31	21, 30	syl5ibrcom 249	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜃)))
32		tfinds.3	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
33	32	biimprd 250	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
34	33	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 = suc 𝑦 → (𝜃 → 𝜑)))
35	31, 34	syldd 72	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
36	15, 35	rexlimi 3264	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
37	12, 36	jaoi 868	. . . . . 6 ⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
38	8, 37	syl6 35	. . . . 5 ⊢ (𝑥 ∈ On → ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
39	5, 38	biimtrrid 245	. . . 4 ⊢ (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
40	4, 39	biimtrid 244	. . 3 ⊢ (𝑥 ∈ On → (¬ Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
41		tfinds.7	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
42	40, 41	pm2.61d2 182	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
43	1, 2, 42	tfis3 7840	1 ⊢ (𝐴 ∈ On → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562  [wsb 2092   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  ∅c0 4287  Ord word 6347  Oncon0 6348  Lim wlim 6349  suc csuc 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354

This theorem is referenced by:  tfindsg  7843  tfindes  7845  tfinds3  7847  oa0r  8509  om0r  8510  om1r  8514  oe1m  8516  oeoalem  8568  r1sdom  9734  r1tr  9736  alephon  10027  alephcard  10028  alephordi  10032  constrsscn  34039  constr01  34041  constrmon  34043  constrconj  34044  rdgprc  36147