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Theorem tfis 4249
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]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 3019 . . . . 5 {x On ∣ φ} ⊆ On
2 nfcv 2175 . . . . . . 7 xz
3 nfrab1 2483 . . . . . . . . 9 x{x On ∣ φ}
42, 3nfss 2932 . . . . . . . 8 x z ⊆ {x On ∣ φ}
53nfcri 2169 . . . . . . . 8 x z {x On ∣ φ}
64, 5nfim 1461 . . . . . . 7 x(z ⊆ {x On ∣ φ} → z {x On ∣ φ})
7 dfss3 2929 . . . . . . . . 9 (x ⊆ {x On ∣ φ} ↔ y x y {x On ∣ φ})
8 sseq1 2960 . . . . . . . . 9 (x = z → (x ⊆ {x On ∣ φ} ↔ z ⊆ {x On ∣ φ}))
97, 8syl5bbr 183 . . . . . . . 8 (x = z → (y x y {x On ∣ φ} ↔ z ⊆ {x On ∣ φ}))
10 rabid 2479 . . . . . . . . 9 (x {x On ∣ φ} ↔ (x On φ))
11 eleq1 2097 . . . . . . . . 9 (x = z → (x {x On ∣ φ} ↔ z {x On ∣ φ}))
1210, 11syl5bbr 183 . . . . . . . 8 (x = z → ((x On φ) ↔ z {x On ∣ φ}))
139, 12imbi12d 223 . . . . . . 7 (x = z → ((y x y {x On ∣ φ} → (x On φ)) ↔ (z ⊆ {x On ∣ φ} → z {x On ∣ φ})))
14 sbequ 1718 . . . . . . . . . . . 12 (w = y → ([w / x]φ ↔ [y / x]φ))
15 nfcv 2175 . . . . . . . . . . . . 13 xOn
16 nfcv 2175 . . . . . . . . . . . . 13 wOn
17 nfv 1418 . . . . . . . . . . . . 13 wφ
18 nfs1v 1812 . . . . . . . . . . . . 13 x[w / x]φ
19 sbequ12 1651 . . . . . . . . . . . . 13 (x = w → (φ ↔ [w / x]φ))
2015, 16, 17, 18, 19cbvrab 2549 . . . . . . . . . . . 12 {x On ∣ φ} = {w On ∣ [w / x]φ}
2114, 20elrab2 2694 . . . . . . . . . . 11 (y {x On ∣ φ} ↔ (y On [y / x]φ))
2221simprbi 260 . . . . . . . . . 10 (y {x On ∣ φ} → [y / x]φ)
2322ralimi 2378 . . . . . . . . 9 (y x y {x On ∣ φ} → y x [y / x]φ)
24 tfis.1 . . . . . . . . 9 (x On → (y x [y / x]φφ))
2523, 24syl5 28 . . . . . . . 8 (x On → (y x y {x On ∣ φ} → φ))
2625anc2li 312 . . . . . . 7 (x On → (y x y {x On ∣ φ} → (x On φ)))
272, 6, 13, 26vtoclgaf 2612 . . . . . 6 (z On → (z ⊆ {x On ∣ φ} → z {x On ∣ φ}))
2827rgen 2368 . . . . 5 z On (z ⊆ {x On ∣ φ} → z {x On ∣ φ})
29 tfi 4248 . . . . 5 (({x On ∣ φ} ⊆ On z On (z ⊆ {x On ∣ φ} → z {x On ∣ φ})) → {x On ∣ φ} = On)
301, 28, 29mp2an 402 . . . 4 {x On ∣ φ} = On
3130eqcomi 2041 . . 3 On = {x On ∣ φ}
3231rabeq2i 2548 . 2 (x On ↔ (x On φ))
3332simprbi 260 1 (x On → φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  [wsb 1642  wral 2300  {crab 2304  wss 2911  Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  tfis2f  4250
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