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Theorem tfi 4333
 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 . (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
tfi
Distinct variable group:   ,

Proof of Theorem tfi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ral 2328 . . . . . . 7
2 imdi 243 . . . . . . . 8
32albii 1375 . . . . . . 7
41, 3bitri 177 . . . . . 6
5 dfss2 2962 . . . . . . . . . 10
65imbi2i 219 . . . . . . . . 9
7 19.21v 1769 . . . . . . . . 9
86, 7bitr4i 180 . . . . . . . 8
98imbi1i 231 . . . . . . 7
109albii 1375 . . . . . 6
114, 10bitri 177 . . . . 5
12 simpl 106 . . . . . . . . . . 11
13 tron 4147 . . . . . . . . . . . . . 14
14 dftr2 3884 . . . . . . . . . . . . . 14
1513, 14mpbi 137 . . . . . . . . . . . . 13
1615spi 1445 . . . . . . . . . . . 12
1716spi 1445 . . . . . . . . . . 11
1812, 17jca 294 . . . . . . . . . 10
1918imim1i 58 . . . . . . . . 9
20 impexp 254 . . . . . . . . 9
21 impexp 254 . . . . . . . . . 10
22 bi2.04 241 . . . . . . . . . 10
2321, 22bitri 177 . . . . . . . . 9
2419, 20, 233imtr3i 193 . . . . . . . 8
2524alimi 1360 . . . . . . 7
2625imim1i 58 . . . . . 6
2726alimi 1360 . . . . 5
2811, 27sylbi 118 . . . 4
30 sbim 1843 . . . . . . . . . 10
31 clelsb3 2158 . . . . . . . . . . 11
32 clelsb3 2158 . . . . . . . . . . 11
3331, 32imbi12i 232 . . . . . . . . . 10
3430, 33bitri 177 . . . . . . . . 9
3534ralbii 2347 . . . . . . . 8
36 df-ral 2328 . . . . . . . 8
3735, 36bitri 177 . . . . . . 7
3837imbi1i 231 . . . . . 6
3938albii 1375 . . . . 5
40 ax-setind 4290 . . . . 5
4139, 40sylbir 129 . . . 4
42 dfss2 2962 . . . 4
4341, 42sylibr 141 . . 3
4429, 43syl 14 . 2
45 eqss 2988 . . 3
4645biimpri 128 . 2
4744, 46syldan 270 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101  wal 1257   wceq 1259   wcel 1409  wsb 1661  wral 2323   wss 2945   wtr 3882  con0 4128 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133 This theorem is referenced by:  tfis  4334
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