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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 4915 | . . . 4 | |
2 | cotr 4915 | . . . . . 6 | |
3 | brin 3975 | . . . . . . . . . . . . 13 | |
4 | brin 3975 | . . . . . . . . . . . . 13 | |
5 | simpr 109 | . . . . . . . . . . . . . . . 16 | |
6 | simpl 108 | . . . . . . . . . . . . . . . 16 | |
7 | 5, 6 | anim12d 333 | . . . . . . . . . . . . . . 15 |
8 | 7 | com12 30 | . . . . . . . . . . . . . 14 |
9 | 8 | an4s 577 | . . . . . . . . . . . . 13 |
10 | 3, 4, 9 | syl2anb 289 | . . . . . . . . . . . 12 |
11 | 10 | com12 30 | . . . . . . . . . . 11 |
12 | brin 3975 | . . . . . . . . . . 11 | |
13 | 11, 12 | syl6ibr 161 | . . . . . . . . . 10 |
14 | 13 | alanimi 1435 | . . . . . . . . 9 |
15 | 14 | alanimi 1435 | . . . . . . . 8 |
16 | 15 | alanimi 1435 | . . . . . . 7 |
17 | 16 | ex 114 | . . . . . 6 |
18 | 2, 17 | sylbi 120 | . . . . 5 |
19 | 18 | com12 30 | . . . 4 |
20 | 1, 19 | sylbi 120 | . . 3 |
21 | 20 | imp 123 | . 2 |
22 | cotr 4915 | . 2 | |
23 | 21, 22 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 cin 3065 wss 3066 class class class wbr 3924 ccom 4538 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-co 4543 |
This theorem is referenced by: trinxp 4927 |
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