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Mirrors > Home > ILE Home > Th. List > relcnvtr | Unicode version |
Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
relcnvtr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 4719 | . . 3 | |
2 | cnvss 4707 | . . 3 | |
3 | 1, 2 | eqsstrrid 3139 | . 2 |
4 | cnvco 4719 | . . . 4 | |
5 | cnvss 4707 | . . . 4 | |
6 | sseq1 3115 | . . . . 5 | |
7 | dfrel2 4984 | . . . . . . 7 | |
8 | coeq1 4691 | . . . . . . . . . 10 | |
9 | coeq2 4692 | . . . . . . . . . 10 | |
10 | 8, 9 | eqtrd 2170 | . . . . . . . . 9 |
11 | id 19 | . . . . . . . . 9 | |
12 | 10, 11 | sseq12d 3123 | . . . . . . . 8 |
13 | 12 | biimpd 143 | . . . . . . 7 |
14 | 7, 13 | sylbi 120 | . . . . . 6 |
15 | 14 | com12 30 | . . . . 5 |
16 | 6, 15 | syl6bi 162 | . . . 4 |
17 | 4, 5, 16 | mpsyl 65 | . . 3 |
18 | 17 | com12 30 | . 2 |
19 | 3, 18 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wss 3066 ccnv 4533 ccom 4538 wrel 4539 |
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-cnv 4542 df-co 4543 |
This theorem is referenced by: (None) |
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