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Theorem tfis 4594
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 3252 . . . . 5 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfcv 2329 . . . . . . 7 𝑥𝑧
3 nfrab1 2667 . . . . . . . . 9 𝑥{𝑥 ∈ On ∣ 𝜑}
42, 3nfss 3160 . . . . . . . 8 𝑥 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}
53nfcri 2323 . . . . . . . 8 𝑥 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}
64, 5nfim 1582 . . . . . . 7 𝑥(𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
7 dfss3 3157 . . . . . . . . 9 (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ ∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
8 sseq1 3190 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ⊆ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
97, 8bitr3id 194 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑}))
10 rabid 2663 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
11 eleq1 2250 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
1210, 11bitr3id 194 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑥 ∈ On ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
139, 12imbi12d 234 . . . . . . 7 (𝑥 = 𝑧 → ((∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)) ↔ (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})))
14 sbequ 1850 . . . . . . . . . . . 12 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 nfcv 2329 . . . . . . . . . . . . 13 𝑥On
16 nfcv 2329 . . . . . . . . . . . . 13 𝑤On
17 nfv 1538 . . . . . . . . . . . . 13 𝑤𝜑
18 nfs1v 1949 . . . . . . . . . . . . 13 𝑥[𝑤 / 𝑥]𝜑
19 sbequ12 1781 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
2015, 16, 17, 18, 19cbvrab 2747 . . . . . . . . . . . 12 {𝑥 ∈ On ∣ 𝜑} = {𝑤 ∈ On ∣ [𝑤 / 𝑥]𝜑}
2114, 20elrab2 2908 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑦 ∈ On ∧ [𝑦 / 𝑥]𝜑))
2221simprbi 275 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → [𝑦 / 𝑥]𝜑)
2322ralimi 2550 . . . . . . . . 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 329 . . . . . . 7 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} → (𝑥 ∈ On ∧ 𝜑)))
272, 6, 13, 26vtoclgaf 2814 . . . . . 6 (𝑧 ∈ On → (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑}))
2827rgen 2540 . . . . 5 𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})
29 tfi 4593 . . . . 5 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∀𝑧 ∈ On (𝑧 ⊆ {𝑥 ∈ On ∣ 𝜑} → 𝑧 ∈ {𝑥 ∈ On ∣ 𝜑})) → {𝑥 ∈ On ∣ 𝜑} = On)
301, 28, 29mp2an 426 . . . 4 {𝑥 ∈ On ∣ 𝜑} = On
3130eqcomi 2191 . . 3 On = {𝑥 ∈ On ∣ 𝜑}
3231rabeq2i 2746 . 2 (𝑥 ∈ On ↔ (𝑥 ∈ On ∧ 𝜑))
3332simprbi 275 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  [wsb 1772  wcel 2158  wral 2465  {crab 2469  wss 3141  Oncon0 4375
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380
This theorem is referenced by:  tfis2f  4595
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