Theorem tfis 7843

Description: Transfinite Induction Schema. If all ordinal numbers less than a given number 𝑥 have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)

Hypothesis

Ref	Expression
tfis.1	⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))

Assertion

Ref	Expression
tfis	⊢ (𝑥 ∈ On → 𝜑)

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfis

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4077	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
3		nfrab1 3451	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
4	2, 3	nfss 3974	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
5	3	nfcri 2890	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
6	4, 5	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7		dfss3 3970	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8		sseq1 4007	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
9	7, 8	bitr3id 284	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10		rabid 3452	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11		eleq1w 2816	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
12	10, 11	bitr3id 284	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
13	9, 12	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14		sbequ 2086	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥On
16		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤On
17		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤𝜑
18		nfs1v 2153	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
19		sbequ12 2243	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
20	15, 16, 17, 18, 19	cbvrabw 3467	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
21	14, 20	elrab2 3686	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
22	21	simprbi 497	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
23	22	ralimi 3083	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑)
24		tfis.1	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))
25	23, 24	syl5 34	. . . . . . . 8 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜑))
26	25	anc2li 556	. . . . . . 7 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
27	2, 6, 13, 26	vtoclgaf 3564	. . . . . 6 ⊢ (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
28	27	rgen 3063	. . . . 5 ⊢ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29		tfi 7841	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
30	1, 28, 29	mp2an 690	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} = On
31	30	eqcomi 2741	. . 3 ⊢ On = {𝑥 ∈ On ∣ 𝜑}
32	31	reqabi 3454	. 2 ⊢ (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
33	32	simprbi 497	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  [wsb 2067   ∈ wcel 2106  ∀wral 3061  {crab 3432   ⊆ wss 3948  Oncon0 6364

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  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 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368

This theorem is referenced by:  tfis2f  7844