Theorem tfinds 7576

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. (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 7564	. . . . 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 404	. . . . 5 ⊢ ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6		eloni 6203	. . . . . . 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 3221	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑥 𝜒
14		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
15	13, 14	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝑥 𝜒 → 𝜑)
16		vex 3499	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	16	sucid 6272	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ suc 𝑦
18	1	rspcv 3620	. . . . . . . . . . . 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 3407	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝜒
24	23, 1	sbiev 2330	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
25		sbequ 2090	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
26	24, 25	syl5bbr 287	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
27	26	cbvralvw 3451	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
28		cbvralsvw 3469	. . . . . . . . . . . 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 344	. . . . . . . . . 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 3317	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
37	12, 36	jaoi 853	. . . . . 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	syl5bir 245	. . . 4 ⊢ (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
40	4, 39	syl5bi 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 183	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
43	1, 2, 42	tfis3 7574	1 ⊢ (𝐴 ∈ On → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  [wsb 2069   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ∅c0 4293  Ord word 6192  Oncon0 6193  Lim wlim 6194  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199

This theorem is referenced by:  tfindsg  7577  tfindes  7579  tfinds3  7581  oa0r  8165  om0r  8166  om1r  8171  oe1m  8173  oeoalem  8224  r1sdom  9205  r1tr  9207  alephon  9497  alephcard  9498  alephordi  9502  rdgprc  33041