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Theorem tfis 7844
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 4078 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfcv 2904 . . . . . . 7 𝑥𝑧
3 nfrab1 3452 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝜑}
42, 3nfss 3975 . . . . . . . 8 𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
53nfcri 2891 . . . . . . . 8 𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
64, 5nfim 1900 . . . . . . 7 𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7 dfss3 3971 . . . . . . . . 9 (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8 sseq1 4008 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
97, 8bitr3id 285 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10 rabid 3453 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11 eleq1w 2817 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
1210, 11bitr3id 285 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
139, 12imbi12d 345 . . . . . . 7 (𝑥 = 𝑧 → ((∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14 sbequ 2087 . . . . . . . . . . . 12 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 nfcv 2904 . . . . . . . . . . . . 13 𝑥On
16 nfcv 2904 . . . . . . . . . . . . 13 𝑤On
17 nfv 1918 . . . . . . . . . . . . 13 𝑤𝜑
18 nfs1v 2154 . . . . . . . . . . . . 13 𝑥[𝑤 / 𝑥]𝜑
19 sbequ12 2244 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
2015, 16, 17, 18, 19cbvrabw 3468 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
2114, 20elrab2 3687 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
2221simprbi 498 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
2322ralimi 3084 . . . . . . . . 9 (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑)
24 tfis.1 . . . . . . . . 9 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
2523, 24syl5 34 . . . . . . . 8 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜑))
2625anc2li 557 . . . . . . 7 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
272, 6, 13, 26vtoclgaf 3565 . . . . . 6 (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2827rgen 3064 . . . . 5 𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29 tfi 7842 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
301, 28, 29mp2an 691 . . . 4 {𝑥 ∈ On ∣ 𝜑} = On
3130eqcomi 2742 . . 3 On = {𝑥 ∈ On ∣ 𝜑}
3231reqabi 3455 . 2 (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
3332simprbi 498 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  [wsb 2068  wcel 2107  wral 3062  {crab 3433  wss 3949  Oncon0 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  tfis2f  7845
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