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Mirrors > Home > ILE Home > Th. List > sotritrieq | Unicode version |
Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.) |
Ref | Expression |
---|---|
sotritric.or | |
sotritric.tri |
Ref | Expression |
---|---|
sotritrieq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sotritric.or | . . . . . . 7 | |
2 | sonr 4072 | . . . . . . 7 | |
3 | 1, 2 | mpan 414 | . . . . . 6 |
4 | breq2 3789 | . . . . . . 7 | |
5 | 4 | notbid 624 | . . . . . 6 |
6 | 3, 5 | syl5ibcom 153 | . . . . 5 |
7 | breq1 3788 | . . . . . . 7 | |
8 | 7 | notbid 624 | . . . . . 6 |
9 | 3, 8 | syl5ibcom 153 | . . . . 5 |
10 | 6, 9 | jcad 301 | . . . 4 |
11 | ioran 701 | . . . 4 | |
12 | 10, 11 | syl6ibr 160 | . . 3 |
13 | 12 | adantr 270 | . 2 |
14 | sotritric.tri | . . 3 | |
15 | 3orrot 925 | . . . . . . 7 | |
16 | 3orcomb 928 | . . . . . . 7 | |
17 | 3orass 922 | . . . . . . 7 | |
18 | 15, 16, 17 | 3bitri 204 | . . . . . 6 |
19 | 18 | biimpi 118 | . . . . 5 |
20 | 19 | orcomd 680 | . . . 4 |
21 | 20 | ord 675 | . . 3 |
22 | 14, 21 | syl 14 | . 2 |
23 | 13, 22 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3o 918 wceq 1284 wcel 1433 class class class wbr 3785 wor 4050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-po 4051 df-iso 4052 |
This theorem is referenced by: distrlem4prl 6774 distrlem4pru 6775 |
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