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Theorem tfis 4373
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 3095 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfcv 2225 . . . . . . 7 𝑥𝑧
3 nfrab1 2542 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝜑}
42, 3nfss 3007 . . . . . . . 8 𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
53nfcri 2219 . . . . . . . 8 𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
64, 5nfim 1507 . . . . . . 7 𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7 dfss3 3004 . . . . . . . . 9 (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8 sseq1 3036 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
97, 8syl5bbr 192 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10 rabid 2538 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11 eleq1 2147 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
1210, 11syl5bbr 192 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
139, 12imbi12d 232 . . . . . . 7 (𝑥 = 𝑧 → ((∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14 sbequ 1765 . . . . . . . . . . . 12 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 nfcv 2225 . . . . . . . . . . . . 13 𝑥On
16 nfcv 2225 . . . . . . . . . . . . 13 𝑤On
17 nfv 1464 . . . . . . . . . . . . 13 𝑤𝜑
18 nfs1v 1860 . . . . . . . . . . . . 13 𝑥[𝑤 / 𝑥]𝜑
19 sbequ12 1698 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
2015, 16, 17, 18, 19cbvrab 2613 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
2114, 20elrab2 2765 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
2221simprbi 269 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
2322ralimi 2434 . . . . . . . . 9 (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → ∀𝑦𝑥 [𝑦 / 𝑥]𝜑)
24 tfis.1 . . . . . . . . 9 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
2523, 24syl5 32 . . . . . . . 8 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → 𝜑))
2625anc2li 322 . . . . . . 7 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
272, 6, 13, 26vtoclgaf 2677 . . . . . 6 (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2827rgen 2424 . . . . 5 𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29 tfi 4372 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
301, 28, 29mp2an 417 . . . 4 {𝑥 ∈ On ∣ 𝜑} = On
3130eqcomi 2089 . . 3 On = {𝑥 ∈ On ∣ 𝜑}
3231rabeq2i 2612 . 2 (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
3332simprbi 269 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  [wsb 1689  wral 2355  {crab 2359  wss 2988  Oncon0 4166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-setind 4328
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-uni 3639  df-tr 3914  df-iord 4169  df-on 4171
This theorem is referenced by:  tfis2f  4374
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