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Mirrors > Home > ILE Home > Th. List > tfi | Unicode version |
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.) |
Ref | Expression |
---|---|
tfi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2453 | . . . . . . 7 | |
2 | imdi 249 | . . . . . . . 8 | |
3 | 2 | albii 1463 | . . . . . . 7 |
4 | 1, 3 | bitri 183 | . . . . . 6 |
5 | dfss2 3136 | . . . . . . . . . 10 | |
6 | 5 | imbi2i 225 | . . . . . . . . 9 |
7 | 19.21v 1866 | . . . . . . . . 9 | |
8 | 6, 7 | bitr4i 186 | . . . . . . . 8 |
9 | 8 | imbi1i 237 | . . . . . . 7 |
10 | 9 | albii 1463 | . . . . . 6 |
11 | 4, 10 | bitri 183 | . . . . 5 |
12 | simpl 108 | . . . . . . . . . . 11 | |
13 | tron 4367 | . . . . . . . . . . . . . 14 | |
14 | dftr2 4089 | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | mpbi 144 | . . . . . . . . . . . . 13 |
16 | 15 | spi 1529 | . . . . . . . . . . . 12 |
17 | 16 | spi 1529 | . . . . . . . . . . 11 |
18 | 12, 17 | jca 304 | . . . . . . . . . 10 |
19 | 18 | imim1i 60 | . . . . . . . . 9 |
20 | impexp 261 | . . . . . . . . 9 | |
21 | impexp 261 | . . . . . . . . . 10 | |
22 | bi2.04 247 | . . . . . . . . . 10 | |
23 | 21, 22 | bitri 183 | . . . . . . . . 9 |
24 | 19, 20, 23 | 3imtr3i 199 | . . . . . . . 8 |
25 | 24 | alimi 1448 | . . . . . . 7 |
26 | 25 | imim1i 60 | . . . . . 6 |
27 | 26 | alimi 1448 | . . . . 5 |
28 | 11, 27 | sylbi 120 | . . . 4 |
29 | 28 | adantl 275 | . . 3 |
30 | sbim 1946 | . . . . . . . . . 10 | |
31 | clelsb1 2275 | . . . . . . . . . . 11 | |
32 | clelsb1 2275 | . . . . . . . . . . 11 | |
33 | 31, 32 | imbi12i 238 | . . . . . . . . . 10 |
34 | 30, 33 | bitri 183 | . . . . . . . . 9 |
35 | 34 | ralbii 2476 | . . . . . . . 8 |
36 | df-ral 2453 | . . . . . . . 8 | |
37 | 35, 36 | bitri 183 | . . . . . . 7 |
38 | 37 | imbi1i 237 | . . . . . 6 |
39 | 38 | albii 1463 | . . . . 5 |
40 | ax-setind 4521 | . . . . 5 | |
41 | 39, 40 | sylbir 134 | . . . 4 |
42 | dfss2 3136 | . . . 4 | |
43 | 41, 42 | sylibr 133 | . . 3 |
44 | 29, 43 | syl 14 | . 2 |
45 | eqss 3162 | . . 3 | |
46 | 45 | biimpri 132 | . 2 |
47 | 44, 46 | syldan 280 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wceq 1348 wsb 1755 wcel 2141 wral 2448 wss 3121 wtr 4087 con0 4348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 |
This theorem is referenced by: tfis 4567 |
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