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Theorem tfis 4834
 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 3428 . . . . 5
2 nfcv 2572 . . . . . . 7
3 nfrab1 2888 . . . . . . . . 9
42, 3nfss 3341 . . . . . . . 8
53nfcri 2566 . . . . . . . 8
64, 5nfim 1832 . . . . . . 7
7 dfss3 3338 . . . . . . . . 9
8 sseq1 3369 . . . . . . . . 9
97, 8syl5bbr 251 . . . . . . . 8
10 rabid 2884 . . . . . . . . 9
11 eleq1 2496 . . . . . . . . 9
1210, 11syl5bbr 251 . . . . . . . 8
139, 12imbi12d 312 . . . . . . 7
14 sbequ 2111 . . . . . . . . . . . 12
15 nfcv 2572 . . . . . . . . . . . . 13
16 nfcv 2572 . . . . . . . . . . . . 13
17 nfv 1629 . . . . . . . . . . . . 13
18 nfs1v 2182 . . . . . . . . . . . . 13
19 sbequ12 1944 . . . . . . . . . . . . 13
2015, 16, 17, 18, 19cbvrab 2954 . . . . . . . . . . . 12
2114, 20elrab2 3094 . . . . . . . . . . 11
2221simprbi 451 . . . . . . . . . 10
2322ralimi 2781 . . . . . . . . 9
24 tfis.1 . . . . . . . . 9
2523, 24syl5 30 . . . . . . . 8
2625anc2li 541 . . . . . . 7
272, 6, 13, 26vtoclgaf 3016 . . . . . 6
2827rgen 2771 . . . . 5
29 tfi 4833 . . . . 5
301, 28, 29mp2an 654 . . . 4
3130eqcomi 2440 . . 3
3231rabeq2i 2953 . 2
3332simprbi 451 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652  wsb 1658   wcel 1725  wral 2705  crab 2709   wss 3320  con0 4581 This theorem is referenced by:  tfis2f  4835 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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