Theorem tfinds 7783

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 7770	. . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
4	3	notbii 320	. . . 4 ⊢ (¬ Lim 𝑥 ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
5		iman 403	. . . . 5 ⊢ ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6		eloni 6320	. . . . . . 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 258	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
12	11	a1d 25	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
13		nfra1 3265	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑥 𝜒
14		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
15	13, 14	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝑥 𝜒 → 𝜑)
16		vex 3447	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	16	sucid 6392	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ suc 𝑦
18	1	rspcv 3572	. . . . . . . . . . . 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 3307	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝜒
24	23, 1	sbiev 2309	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
25		sbequ 2086	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
26	24, 25	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
27	26	cbvralvw 3223	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
28		cbvralsvw 3298	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ suc 𝑦𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑)
29	22, 27, 28	3bitr4g 314	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑥 ∈ suc 𝑦𝜑))
30	29	imbi1d 342	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → ((∀𝑦 ∈ 𝑥 𝜒 → 𝜃) ↔ (∀𝑥 ∈ suc 𝑦𝜑 → 𝜃)))
31	21, 30	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜃)))
32		tfinds.3	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
33	32	biimprd 248	. . . . . . . . . 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 3240	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
37	12, 36	jaoi 855	. . . . . 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 242	. . . 4 ⊢ (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
40	4, 39	biimtrid 241	. . 3 ⊢ (𝑥 ∈ On → (¬ Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
41		tfinds.7	. . 3 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
42	40, 41	pm2.61d2 181	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
43	1, 2, 42	tfis3 7781	1 ⊢ (𝐴 ∈ On → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 845   = wceq 1541  [wsb 2067   ∈ wcel 2106  ∀wral 3062  ∃wrex 3071  ∅c0 4277  Ord word 6309  Oncon0 6310  Lim wlim 6311  suc csuc 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379  ax-un 7659

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3924  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-br 5101  df-opab 5163  df-tr 5218  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5582  df-we 5584  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316

This theorem is referenced by:  tfindsg  7784  tfindes  7786  tfinds3  7788  oa0r  8448  om0r  8449  om1r  8454  oe1m  8456  oeoalem  8507  r1sdom  9640  r1tr  9642  alephon  9935  alephcard  9936  alephordi  9940  rdgprc  34117