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Mirrors > Home > ILE Home > Th. List > trin2 | Unicode version |
Description: The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.) |
Ref | Expression |
---|---|
trin2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cotr 4706 | . . . 4 | |
2 | cotr 4706 | . . . . . 6 | |
3 | brin 3811 | . . . . . . . . . . . . 13 | |
4 | brin 3811 | . . . . . . . . . . . . 13 | |
5 | simpr 103 | . . . . . . . . . . . . . . . 16 | |
6 | simpl 102 | . . . . . . . . . . . . . . . 16 | |
7 | 5, 6 | anim12d 318 | . . . . . . . . . . . . . . 15 |
8 | 7 | com12 27 | . . . . . . . . . . . . . 14 |
9 | 8 | an4s 522 | . . . . . . . . . . . . 13 |
10 | 3, 4, 9 | syl2anb 275 | . . . . . . . . . . . 12 |
11 | 10 | com12 27 | . . . . . . . . . . 11 |
12 | brin 3811 | . . . . . . . . . . 11 | |
13 | 11, 12 | syl6ibr 151 | . . . . . . . . . 10 |
14 | 13 | alanimi 1348 | . . . . . . . . 9 |
15 | 14 | alanimi 1348 | . . . . . . . 8 |
16 | 15 | alanimi 1348 | . . . . . . 7 |
17 | 16 | ex 108 | . . . . . 6 |
18 | 2, 17 | sylbi 114 | . . . . 5 |
19 | 18 | com12 27 | . . . 4 |
20 | 1, 19 | sylbi 114 | . . 3 |
21 | 20 | imp 115 | . 2 |
22 | cotr 4706 | . 2 | |
23 | 21, 22 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 cin 2916 wss 2917 class class class wbr 3764 ccom 4349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-co 4354 |
This theorem is referenced by: trinxp 4718 |
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