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Theorem tfis 6689
 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:   ,   ,

Proof of Theorem tfis
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3584 . . . . 5
2 nfcv 2619 . . . . . . 7
3 nfrab1 3038 . . . . . . . . 9
42, 3nfss 3496 . . . . . . . 8
53nfcri 2612 . . . . . . . 8
64, 5nfim 1920 . . . . . . 7
7 dfss3 3493 . . . . . . . . 9
8 sseq1 3524 . . . . . . . . 9
97, 8syl5bbr 259 . . . . . . . 8
10 rabid 3034 . . . . . . . . 9
11 eleq1 2529 . . . . . . . . 9
1210, 11syl5bbr 259 . . . . . . . 8
139, 12imbi12d 320 . . . . . . 7
14 sbequ 2117 . . . . . . . . . . . 12
15 nfcv 2619 . . . . . . . . . . . . 13
16 nfcv 2619 . . . . . . . . . . . . 13
17 nfv 1707 . . . . . . . . . . . . 13
18 nfs1v 2181 . . . . . . . . . . . . 13
19 sbequ12 1992 . . . . . . . . . . . . 13
2015, 16, 17, 18, 19cbvrab 3107 . . . . . . . . . . . 12
2114, 20elrab2 3259 . . . . . . . . . . 11
2221simprbi 464 . . . . . . . . . 10
2322ralimi 2850 . . . . . . . . 9
24 tfis.1 . . . . . . . . 9
2523, 24syl5 32 . . . . . . . 8
2625anc2li 557 . . . . . . 7
272, 6, 13, 26vtoclgaf 3172 . . . . . 6
2827rgen 2817 . . . . 5
29 tfi 6688 . . . . 5
301, 28, 29mp2an 672 . . . 4
3130eqcomi 2470 . . 3
3231rabeq2i 3106 . 2
3332simprbi 464 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  A.wral 2807  {crab 2811  C_wss 3475   con0 4883 This theorem is referenced by:  tfis2f  6690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887
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