Theorem tfinds 7804

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 7791	. . . . 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 401	. . . . 5 ⊢ ((Ord 𝑥 → (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) ↔ ¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)))
6		eloni 6328	. . . . . . 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 3261	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝑥 𝜒
14		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
15	13, 14	nfim 1898	. . . . . . . 8 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝑥 𝜒 → 𝜑)
16		vex 3445	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	16	sucid 6402	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ suc 𝑦
18	1	rspcv 3573	. . . . . . . . . . . 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 3294	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ suc 𝑦[𝑧 / 𝑥]𝜑))
23		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝜒
24	23, 1	sbiev 2320	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
25		sbequ 2089	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
26	24, 25	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝜒 ↔ [𝑧 / 𝑥]𝜑))
27	26	cbvralvw 3215	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 𝜒 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
28		cbvralsvw 3288	. . . . . . . . . . . 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 341	. . . . . . . . . 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 3237	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 = suc 𝑦 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))
37	12, 36	jaoi 858	. . . . . 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 243	. . . 4 ⊢ (𝑥 ∈ On → (¬ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)) → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
40	4, 39	biimtrid 242	. . 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 7802	1 ⊢ (𝐴 ∈ On → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  [wsb 2068   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  ∅c0 4286  Ord word 6317  Oncon0 6318  Lim wlim 6319  suc csuc 6320

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324

This theorem is referenced by:  tfindsg  7805  tfindes  7807  tfinds3  7809  oa0r  8467  om0r  8468  om1r  8472  oe1m  8474  oeoalem  8526  r1sdom  9690  r1tr  9692  alephon  9983  alephcard  9984  alephordi  9988  constrsscn  33899  constr01  33901  constrmon  33903  constrconj  33904  rdgprc  35988