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Theorem tfi 4833
 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 4842 or tfinds 4839 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 3470 . . . . . . . . 9
21adantl 453 . . . . . . . 8
3 eldifi 3469 . . . . . . . . . 10
4 onss 4771 . . . . . . . . . . . . 13
5 difin0ss 3694 . . . . . . . . . . . . 13
64, 5syl5com 28 . . . . . . . . . . . 12
76imim1d 71 . . . . . . . . . . 11
87a2i 13 . . . . . . . . . 10
93, 8syl5 30 . . . . . . . . 9
109imp 419 . . . . . . . 8
112, 10mtod 170 . . . . . . 7
1211ex 424 . . . . . 6
1312ralimi2 2778 . . . . 5
14 ralnex 2715 . . . . 5
1513, 14sylib 189 . . . 4
16 ssdif0 3686 . . . . . 6
1716necon3bbii 2632 . . . . 5
18 ordon 4763 . . . . . 6
19 difss 3474 . . . . . 6
20 tz7.5 4602 . . . . . 6
2118, 19, 20mp3an12 1269 . . . . 5
2217, 21sylbi 188 . . . 4
2315, 22nsyl2 121 . . 3
2423anim2i 553 . 2
25 eqss 3363 . 2
2624, 25sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725   wne 2599  wral 2705  wrex 2706   cdif 3317   cin 3319   wss 3320  c0 3628   word 4580  con0 4581 This theorem is referenced by:  tfis  4834  tfisg  25479 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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