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Theorem tfis 4326
 Description: Transfinite Induction Schema. If all ordinal numbers less than a given number 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
Assertion
Ref Expression
tfis
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem tfis
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3080 . . . . 5
2 nfcv 2220 . . . . . . 7
3 nfrab1 2534 . . . . . . . . 9
42, 3nfss 2993 . . . . . . . 8
53nfcri 2214 . . . . . . . 8
64, 5nfim 1505 . . . . . . 7
7 dfss3 2990 . . . . . . . . 9
8 sseq1 3021 . . . . . . . . 9
97, 8syl5bbr 192 . . . . . . . 8
10 rabid 2530 . . . . . . . . 9
11 eleq1 2142 . . . . . . . . 9
1210, 11syl5bbr 192 . . . . . . . 8
139, 12imbi12d 232 . . . . . . 7
14 sbequ 1762 . . . . . . . . . . . 12
15 nfcv 2220 . . . . . . . . . . . . 13
16 nfcv 2220 . . . . . . . . . . . . 13
17 nfv 1462 . . . . . . . . . . . . 13
18 nfs1v 1857 . . . . . . . . . . . . 13
19 sbequ12 1695 . . . . . . . . . . . . 13
2015, 16, 17, 18, 19cbvrab 2600 . . . . . . . . . . . 12
2114, 20elrab2 2752 . . . . . . . . . . 11
2221simprbi 269 . . . . . . . . . 10
2322ralimi 2427 . . . . . . . . 9
24 tfis.1 . . . . . . . . 9
2523, 24syl5 32 . . . . . . . 8
2625anc2li 322 . . . . . . 7
272, 6, 13, 26vtoclgaf 2664 . . . . . 6
2827rgen 2417 . . . . 5
29 tfi 4325 . . . . 5
301, 28, 29mp2an 417 . . . 4
3130eqcomi 2086 . . 3
3231rabeq2i 2599 . 2
3332simprbi 269 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wceq 1285   wcel 1434  wsb 1686  wral 2349  crab 2353   wss 2974  con0 4120 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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125 This theorem is referenced by:  tfis2f  4327
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