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Theorem tfis 4249
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  On
Assertion
Ref Expression
tfis  On
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 3019 . . . . 5  {  On  |  }  C_  On
2 nfcv 2175 . . . . . . 7  F/_
3 nfrab1 2483 . . . . . . . . 9  F/_ {  On  |  }
42, 3nfss 2932 . . . . . . . 8  F/  C_  {  On  |  }
53nfcri 2169 . . . . . . . 8  F/  {  On  |  }
64, 5nfim 1461 . . . . . . 7  F/  C_  {  On  |  }  {  On  |  }
7 dfss3 2929 . . . . . . . . 9 
C_  {  On  |  }  {  On  |  }
8 sseq1 2960 . . . . . . . . 9  C_  {  On  |  }  C_  {  On  |  }
97, 8syl5bbr 183 . . . . . . . 8  {  On  |  }  C_  {  On  |  }
10 rabid 2479 . . . . . . . . 9  {  On  |  }  On
11 eleq1 2097 . . . . . . . . 9  {  On  |  }  {  On  |  }
1210, 11syl5bbr 183 . . . . . . . 8  On  {  On  |  }
139, 12imbi12d 223 . . . . . . 7  {  On  |  }  On 
C_  {  On  |  }  {  On  |  }
14 sbequ 1718 . . . . . . . . . . . 12
15 nfcv 2175 . . . . . . . . . . . . 13  F/_ On
16 nfcv 2175 . . . . . . . . . . . . 13  F/_ On
17 nfv 1418 . . . . . . . . . . . . 13  F/
18 nfs1v 1812 . . . . . . . . . . . . 13  F/
19 sbequ12 1651 . . . . . . . . . . . . 13
2015, 16, 17, 18, 19cbvrab 2549 . . . . . . . . . . . 12  {  On  |  }  {  On  |  }
2114, 20elrab2 2694 . . . . . . . . . . 11  {  On  |  }  On
2221simprbi 260 . . . . . . . . . 10  {  On  |  }
2322ralimi 2378 . . . . . . . . 9  {  On  |  }
24 tfis.1 . . . . . . . . 9  On
2523, 24syl5 28 . . . . . . . 8  On  {  On  |  }
2625anc2li 312 . . . . . . 7  On  {  On  |  }  On
272, 6, 13, 26vtoclgaf 2612 . . . . . 6  On  C_  {  On  |  }  {  On  |  }
2827rgen 2368 . . . . 5  On  C_  {  On  |  } 
{  On  |  }
29 tfi 4248 . . . . 5  {  On  |  }  C_  On  On  C_  {  On  |  }  {  On  |  }  {  On  |  }  On
301, 28, 29mp2an 402 . . . 4  {  On  |  }  On
3130eqcomi 2041 . . 3  On  {  On  |  }
3231rabeq2i 2548 . 2  On  On
3332simprbi 260 1  On
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wsb 1642  wral 2300   {crab 2304    C_ 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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