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Theorem tfis 6920
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 → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
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.
StepHypRef Expression
1 ssrab2 3646 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfcv 2747 . . . . . . 7 𝑥𝑧
3 nfrab1 3095 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝜑}
42, 3nfss 3557 . . . . . . . 8 𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
53nfcri 2741 . . . . . . . 8 𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
64, 5nfim 1812 . . . . . . 7 𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7 dfss3 3554 . . . . . . . . 9 (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8 sseq1 3585 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
97, 8syl5bbr 272 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10 rabid 3091 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11 eleq1 2672 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
1210, 11syl5bbr 272 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
139, 12imbi12d 332 . . . . . . 7 (𝑥 = 𝑧 → ((∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14 sbequ 2360 . . . . . . . . . . . 12 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 nfcv 2747 . . . . . . . . . . . . 13 𝑥On
16 nfcv 2747 . . . . . . . . . . . . 13 𝑤On
17 nfv 1829 . . . . . . . . . . . . 13 𝑤𝜑
18 nfs1v 2421 . . . . . . . . . . . . 13 𝑥[𝑤 / 𝑥]𝜑
19 sbequ12 2095 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
2015, 16, 17, 18, 19cbvrab 3167 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
2114, 20elrab2 3329 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
2221simprbi 478 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
2322ralimi 2932 . . . . . . . . 9 (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑)
24 tfis.1 . . . . . . . . 9 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
2523, 24syl5 33 . . . . . . . 8 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜑))
2625anc2li 577 . . . . . . 7 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
272, 6, 13, 26vtoclgaf 3240 . . . . . 6 (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2827rgen 2902 . . . . 5 𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29 tfi 6919 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
301, 28, 29mp2an 703 . . . 4 {𝑥 ∈ On ∣ 𝜑} = On
3130eqcomi 2615 . . 3 On = {𝑥 ∈ On ∣ 𝜑}
3231rabeq2i 3166 . 2 (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
3332simprbi 478 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  [wsb 1866  wcel 1976  wral 2892  {crab 2896  wss 3536  Oncon0 5623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-tr 4672  df-eprel 4936  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-ord 5626  df-on 5627
This theorem is referenced by:  tfis2f  6921
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