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Theorem tfindsg2 6696
Description: Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal instead of zero. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)
Hypotheses
Ref Expression
tfindsg2.1
tfindsg2.2
tfindsg2.3
tfindsg2.4
tfindsg2.5
tfindsg2.6
tfindsg2.7
Assertion
Ref Expression
tfindsg2
Distinct variable groups:   ,   , ,   ,   ,   ,   ,

Proof of Theorem tfindsg2
StepHypRef Expression
1 onelon 4908 . . 3
2 sucelon 6652 . . 3
31, 2sylib 196 . 2
4 eloni 4893 . . . 4
5 ordsucss 6653 . . . 4
64, 5syl 16 . . 3
76imp 429 . 2
8 tfindsg2.1 . . . . 5
9 tfindsg2.2 . . . . 5
10 tfindsg2.3 . . . . 5
11 tfindsg2.4 . . . . 5
12 tfindsg2.5 . . . . . 6
132, 12sylbir 213 . . . . 5
14 eloni 4893 . . . . . . . . . 10
15 ordelsuc 6655 . . . . . . . . . 10
1614, 15sylan2 474 . . . . . . . . 9
1716ancoms 453 . . . . . . . 8
18 tfindsg2.6 . . . . . . . . . 10
1918ex 434 . . . . . . . . 9
2019adantr 465 . . . . . . . 8
2117, 20sylbird 235 . . . . . . 7
222, 21sylan2br 476 . . . . . 6
2322imp 429 . . . . 5
24 tfindsg2.7 . . . . . . . . . 10
2524ex 434 . . . . . . . . 9
2625adantr 465 . . . . . . . 8
27 vex 3112 . . . . . . . . . . 11
28 limelon 4946 . . . . . . . . . . 11
2927, 28mpan 670 . . . . . . . . . 10
30 eloni 4893 . . . . . . . . . . . 12
31 ordelsuc 6655 . . . . . . . . . . . 12
3230, 31sylan2 474 . . . . . . . . . . 11
33 onelon 4908 . . . . . . . . . . . . . . . . 17
3433, 14syl 16 . . . . . . . . . . . . . . . 16
3534, 15sylan2 474 . . . . . . . . . . . . . . 15
3635anassrs 648 . . . . . . . . . . . . . 14
3736imbi1d 317 . . . . . . . . . . . . 13
3837ralbidva 2893 . . . . . . . . . . . 12
3938imbi1d 317 . . . . . . . . . . 11
4032, 39imbi12d 320 . . . . . . . . . 10
4129, 40sylan2 474 . . . . . . . . 9
4241ancoms 453 . . . . . . . 8
4326, 42mpbid 210 . . . . . . 7
442, 43sylan2br 476 . . . . . 6
4544imp 429 . . . . 5
468, 9, 10, 11, 13, 23, 45tfindsg 6695 . . . 4
4746expl 618 . . 3
4847adantr 465 . 2
493, 7, 48mp2and 679 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807   cvv 3109  C_wss 3475  Ordword 4882   con0 4883  Limwlim 4884  succsuc 4885
This theorem is referenced by:  oeordi  7255
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-pw 4014  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  df-lim 4888  df-suc 4889
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