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Theorem tfi 6688
Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if is a class of ordinal numbers with the property that every ordinal number included in also belongs to , then every ordinal number is in .

See theorem tfindes 6697 or tfinds 6694 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Assertion
Ref Expression
tfi
Distinct variable group:   ,

Proof of Theorem tfi
StepHypRef Expression
1 eldifn 3626 . . . . . . . . 9
21adantl 466 . . . . . . . 8
3 eldifi 3625 . . . . . . . . . 10
4 onss 6626 . . . . . . . . . . . . 13
5 difin0ss 3894 . . . . . . . . . . . . 13
64, 5syl5com 30 . . . . . . . . . . . 12
76imim1d 75 . . . . . . . . . . 11
87a2i 13 . . . . . . . . . 10
93, 8syl5 32 . . . . . . . . 9
109imp 429 . . . . . . . 8
112, 10mtod 177 . . . . . . 7
1211ex 434 . . . . . 6
1312ralimi2 2847 . . . . 5
14 ralnex 2903 . . . . 5
1513, 14sylib 196 . . . 4
16 ssdif0 3885 . . . . . 6
1716necon3bbii 2718 . . . . 5
18 ordon 6618 . . . . . 6
19 difss 3630 . . . . . 6
20 tz7.5 4904 . . . . . 6
2118, 19, 20mp3an12 1314 . . . . 5
2217, 21sylbi 195 . . . 4
2315, 22nsyl2 127 . . 3
2423anim2i 569 . 2
25 eqss 3518 . 2
2624, 25sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  \cdif 3472  i^icin 3474  C_wss 3475   c0 3784  Ordword 4882   con0 4883
This theorem is referenced by:  tfis  6689  tfisg  29284
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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