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Theorem tfi 4602
 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 4611 or tfinds 4608 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 3260 . . . . . . . . 9
21adantl 454 . . . . . . . 8
3 eldifi 3259 . . . . . . . . . 10
4 onss 4540 . . . . . . . . . . . . 13
5 difin0ss 3481 . . . . . . . . . . . . 13
64, 5syl5com 28 . . . . . . . . . . . 12
76imim1d 71 . . . . . . . . . . 11
87a2i 14 . . . . . . . . . 10
93, 8syl5 30 . . . . . . . . 9
109imp 420 . . . . . . . 8
112, 10mtod 170 . . . . . . 7
1211ex 425 . . . . . 6
1312ralimi2 2588 . . . . 5
14 ralnex 2526 . . . . 5
1513, 14sylib 190 . . . 4
16 ssdif0 3474 . . . . . 6
1716necon3bbii 2450 . . . . 5
18 ordon 4532 . . . . . 6
19 difss 3264 . . . . . 6
20 tz7.5 4371 . . . . . 6
2118, 19, 20mp3an12 1272 . . . . 5
2217, 21sylbi 189 . . . 4
2315, 22nsyl2 121 . . 3
2423anim2i 555 . 2
25 eqss 3155 . 2
2624, 25sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   wceq 1619   wcel 1621   wne 2419  wral 2516  wrex 2517   cdif 3110   cin 3112   wss 3113  c0 3416   word 4349  con0 4350 This theorem is referenced by:  tfis  4603  tfisg  23559 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354
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