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Theorem tfis 4229
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 2998 . . . . 5 {x On ∣ φ} ⊆ On
2 nfcv 2156 . . . . . . 7 xz
3 nfrab1 2463 . . . . . . . . 9 x{x On ∣ φ}
42, 3nfss 2911 . . . . . . . 8 x z ⊆ {x On ∣ φ}
53nfcri 2150 . . . . . . . 8 x z {x On ∣ φ}
64, 5nfim 1442 . . . . . . 7 x(z ⊆ {x On ∣ φ} → z {x On ∣ φ})
7 dfss3 2908 . . . . . . . . 9 (x ⊆ {x On ∣ φ} ↔ y x y {x On ∣ φ})
8 sseq1 2939 . . . . . . . . 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 2459 . . . . . . . . 9 (x {x On ∣ φ} ↔ (x On φ))
11 eleq1 2078 . . . . . . . . 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 1699 . . . . . . . . . . . 12 (w = y → ([w / x]φ ↔ [y / x]φ))
15 nfcv 2156 . . . . . . . . . . . . 13 xOn
16 nfcv 2156 . . . . . . . . . . . . 13 wOn
17 nfv 1398 . . . . . . . . . . . . 13 wφ
18 nfs1v 1793 . . . . . . . . . . . . 13 x[w / x]φ
19 sbequ12 1632 . . . . . . . . . . . . 13 (x = w → (φ ↔ [w / x]φ))
2015, 16, 17, 18, 19cbvrab 2529 . . . . . . . . . . . 12 {x On ∣ φ} = {w On ∣ [w / x]φ}
2114, 20elrab2 2673 . . . . . . . . . . 11 (y {x On ∣ φ} ↔ (y On [y / x]φ))
2221simprbi 260 . . . . . . . . . 10 (y {x On ∣ φ} → [y / x]φ)
2322ralimi 2358 . . . . . . . . 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 2591 . . . . . 6 (z On → (z ⊆ {x On ∣ φ} → z {x On ∣ φ}))
2827rgen 2348 . . . . 5 z On (z ⊆ {x On ∣ φ} → z {x On ∣ φ})
29 tfi 4228 . . . . 5 (({x On ∣ φ} ⊆ On z On (z ⊆ {x On ∣ φ} → z {x On ∣ φ})) → {x On ∣ φ} = On)
301, 28, 29mp2an 404 . . . 4 {x On ∣ φ} = On
3130eqcomi 2022 . . 3 On = {x On ∣ φ}
3231rabeq2i 2528 . 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 1226   wcel 1370  [wsb 1623  wral 2280  {crab 2284  wss 2890  Oncon0 4045
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-in 2897  df-ss 2904  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050
This theorem is referenced by:  tfis2f  4230
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