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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 4209 | . . . . . . 7 | |
3 | 1, 2 | mpan 420 | . . . . . 6 |
4 | breq2 3903 | . . . . . . 7 | |
5 | 4 | notbid 641 | . . . . . 6 |
6 | 3, 5 | syl5ibcom 154 | . . . . 5 |
7 | breq1 3902 | . . . . . . 7 | |
8 | 7 | notbid 641 | . . . . . 6 |
9 | 3, 8 | syl5ibcom 154 | . . . . 5 |
10 | 6, 9 | jcad 305 | . . . 4 |
11 | ioran 726 | . . . 4 | |
12 | 10, 11 | syl6ibr 161 | . . 3 |
13 | 12 | adantr 274 | . 2 |
14 | sotritric.tri | . . 3 | |
15 | 3orrot 953 | . . . . . . 7 | |
16 | 3orcomb 956 | . . . . . . 7 | |
17 | 3orass 950 | . . . . . . 7 | |
18 | 15, 16, 17 | 3bitri 205 | . . . . . 6 |
19 | 18 | biimpi 119 | . . . . 5 |
20 | 19 | orcomd 703 | . . . 4 |
21 | 20 | ord 698 | . . 3 |
22 | 14, 21 | syl 14 | . 2 |
23 | 13, 22 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 class class class wbr 3899 wor 4187 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-po 4188 df-iso 4189 |
This theorem is referenced by: distrlem4prl 7360 distrlem4pru 7361 |
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