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Theorem tfi 4644
 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 4653 or tfinds 4650 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 3299 . . . . . . . . 9
21adantl 452 . . . . . . . 8
3 eldifi 3298 . . . . . . . . . 10
4 onss 4582 . . . . . . . . . . . . 13
5 difin0ss 3520 . . . . . . . . . . . . 13
64, 5syl5com 26 . . . . . . . . . . . 12
76imim1d 69 . . . . . . . . . . 11
87a2i 12 . . . . . . . . . 10
93, 8syl5 28 . . . . . . . . 9
109imp 418 . . . . . . . 8
112, 10mtod 168 . . . . . . 7
1211ex 423 . . . . . 6
1312ralimi2 2615 . . . . 5
14 ralnex 2553 . . . . 5
1513, 14sylib 188 . . . 4
16 ssdif0 3513 . . . . . 6
1716necon3bbii 2477 . . . . 5
18 ordon 4574 . . . . . 6
19 difss 3303 . . . . . 6
20 tz7.5 4413 . . . . . 6
2118, 19, 20mp3an12 1267 . . . . 5
2217, 21sylbi 187 . . . 4
2315, 22nsyl2 119 . . 3
2423anim2i 552 . 2
25 eqss 3194 . 2
2624, 25sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wceq 1623   wcel 1684   wne 2446  wral 2543  wrex 2544   cdif 3149   cin 3151   wss 3152  c0 3455   word 4391  con0 4392 This theorem is referenced by:  tfis  4645  tfisg  24204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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