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Theorem tfis 4541
 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 3213 . . . . 5
2 nfcv 2299 . . . . . . 7
3 nfrab1 2636 . . . . . . . . 9
42, 3nfss 3121 . . . . . . . 8
53nfcri 2293 . . . . . . . 8
64, 5nfim 1552 . . . . . . 7
7 dfss3 3118 . . . . . . . . 9
8 sseq1 3151 . . . . . . . . 9
97, 8bitr3id 193 . . . . . . . 8
10 rabid 2632 . . . . . . . . 9
11 eleq1 2220 . . . . . . . . 9
1210, 11bitr3id 193 . . . . . . . 8
139, 12imbi12d 233 . . . . . . 7
14 sbequ 1820 . . . . . . . . . . . 12
15 nfcv 2299 . . . . . . . . . . . . 13
16 nfcv 2299 . . . . . . . . . . . . 13
17 nfv 1508 . . . . . . . . . . . . 13
18 nfs1v 1919 . . . . . . . . . . . . 13
19 sbequ12 1751 . . . . . . . . . . . . 13
2015, 16, 17, 18, 19cbvrab 2710 . . . . . . . . . . . 12
2114, 20elrab2 2871 . . . . . . . . . . 11
2221simprbi 273 . . . . . . . . . 10
2322ralimi 2520 . . . . . . . . 9
24 tfis.1 . . . . . . . . 9
2523, 24syl5 32 . . . . . . . 8
2625anc2li 327 . . . . . . 7
272, 6, 13, 26vtoclgaf 2777 . . . . . 6
2827rgen 2510 . . . . 5
29 tfi 4540 . . . . 5
301, 28, 29mp2an 423 . . . 4
3130eqcomi 2161 . . 3
3231rabeq2i 2709 . 2
3332simprbi 273 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1335  wsb 1742   wcel 2128  wral 2435  crab 2439   wss 3102  con0 4323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4495 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328 This theorem is referenced by:  tfis2f  4542
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