Theorem tfis 4497

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 3182	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfcv 2281	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
3		nfrab1 2610	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
4	2, 3	nfss 3090	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
5	3	nfcri 2275	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
6	4, 5	nfim 1551	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7		dfss3 3087	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8		sseq1 3120	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
9	7, 8	syl5bbr 193	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10		rabid 2606	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11		eleq1 2202	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
12	10, 11	syl5bbr 193	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
13	9, 12	imbi12d 233	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14		sbequ 1812	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		nfcv 2281	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥On
16		nfcv 2281	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤On
17		nfv 1508	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤𝜑
18		nfs1v 1912	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
19		sbequ12 1744	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
20	15, 16, 17, 18, 19	cbvrab 2684	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
21	14, 20	elrab2 2843	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
22	21	simprbi 273	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
23	22	ralimi 2495	. . . . . . . . 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 32	. . . . . . . 8 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜑))
26	25	anc2li 327	. . . . . . 7 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
27	2, 6, 13, 26	vtoclgaf 2751	. . . . . 6 ⊢ (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
28	27	rgen 2485	. . . . 5 ⊢ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29		tfi 4496	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
30	1, 28, 29	mp2an 422	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} = On
31	30	eqcomi 2143	. . 3 ⊢ On = {𝑥 ∈ On ∣ 𝜑}
32	31	rabeq2i 2683	. 2 ⊢ (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
33	32	simprbi 273	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  [wsb 1735  ∀wral 2416  {crab 2420   ⊆ wss 3071  Oncon0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290

This theorem is referenced by:  tfis2f  4498