Theorem tfis 7561

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 4054	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On
2		nfcv 2975	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
3		nfrab1 3383	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑}
4	2, 3	nfss 3958	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
5	3	nfcri 2969	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
6	4, 5	nfim 1891	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7		dfss3 3954	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8		sseq1 3990	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
9	7, 8	syl5bbr 287	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10		rabid 3377	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11		eleq1w 2893	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
12	10, 11	syl5bbr 287	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
13	9, 12	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14		sbequ 2084	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		nfcv 2975	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥On
16		nfcv 2975	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤On
17		nfv 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤𝜑
18		nfs1v 2267	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
19		sbequ12 2246	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
20	15, 16, 17, 18, 19	cbvrabw 3488	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
21	14, 20	elrab2 3681	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
22	21	simprbi 499	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
23	22	ralimi 3158	. . . . . . . . 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 558	. . . . . . 7 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
27	2, 6, 13, 26	vtoclgaf 3571	. . . . . 6 ⊢ (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
28	27	rgen 3146	. . . . 5 ⊢ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29		tfi 7560	. . . . 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 2828	. . 3 ⊢ On = {𝑥 ∈ On ∣ 𝜑}
32	31	rabeq2i 3486	. 2 ⊢ (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
33	32	simprbi 499	1 ⊢ (𝑥 ∈ On → 𝜑)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531  [wsb 2063   ∈ wcel 2108  ∀wral 3136  {crab 3140   ⊆ wss 3934  Oncon0 6184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188

This theorem is referenced by:  tfis2f  7562