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Mirrors > Home > ILE Home > Th. List > cotr | Unicode version |
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cotr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4594 | . . . 4 | |
2 | 1 | relopabi 4711 | . . 3 |
3 | ssrel 4673 | . . 3 | |
4 | 2, 3 | ax-mp 5 | . 2 |
5 | vex 2715 | . . . . . . . 8 | |
6 | vex 2715 | . . . . . . . 8 | |
7 | 5, 6 | opelco 4757 | . . . . . . 7 |
8 | df-br 3966 | . . . . . . . 8 | |
9 | 8 | bicomi 131 | . . . . . . 7 |
10 | 7, 9 | imbi12i 238 | . . . . . 6 |
11 | 19.23v 1863 | . . . . . 6 | |
12 | 10, 11 | bitr4i 186 | . . . . 5 |
13 | 12 | albii 1450 | . . . 4 |
14 | alcom 1458 | . . . 4 | |
15 | 13, 14 | bitri 183 | . . 3 |
16 | 15 | albii 1450 | . 2 |
17 | 4, 16 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wex 1472 wcel 2128 wss 3102 cop 3563 class class class wbr 3965 ccom 4589 wrel 4590 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4591 df-rel 4592 df-co 4594 |
This theorem is referenced by: xpidtr 4975 trin2 4976 dfer2 6478 |
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