Theorem tfis 4652

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 3289	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfcv 2352	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
3		nfrab1 2691	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
4	2, 3	nfss 3197	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
5	3	nfcri 2346	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
6	4, 5	nfim 1598	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7		dfss3 3193	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8		sseq1 3227	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
9	7, 8	bitr3id 194	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10		rabid 2687	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11		eleq1 2272	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
12	10, 11	bitr3id 194	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
13	9, 12	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14		sbequ 1866	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		nfcv 2352	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥On
16		nfcv 2352	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤On
17		nfv 1554	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤𝜑
18		nfs1v 1970	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
19		sbequ12 1797	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
20	15, 16, 17, 18, 19	cbvrab 2777	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
21	14, 20	elrab2 2942	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
22	21	simprbi 275	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
23	22	ralimi 2573	. . . . . . . . 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 329	. . . . . . 7 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
27	2, 6, 13, 26	vtoclgaf 2846	. . . . . 6 ⊢ (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
28	27	rgen 2563	. . . . 5 ⊢ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29		tfi 4651	. . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
30	1, 28, 29	mp2an 426	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝜑} = On
31	30	eqcomi 2213	. . 3 ⊢ On = {𝑥 ∈ On ∣ 𝜑}
32	31	rabeq2i 2776	. 2 ⊢ (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
33	32	simprbi 275	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375  [wsb 1788   ∈ wcel 2180  ∀wral 2488  {crab 2492   ⊆ wss 3177  Oncon0 4431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-in 3183  df-ss 3190  df-uni 3868  df-tr 4162  df-iord 4434  df-on 4436

This theorem is referenced by:  tfis2f  4653