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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 4300 | . . . . . . 7 | |
3 | 1, 2 | mpan 422 | . . . . . 6 |
4 | breq2 3991 | . . . . . . 7 | |
5 | 4 | notbid 662 | . . . . . 6 |
6 | 3, 5 | syl5ibcom 154 | . . . . 5 |
7 | breq1 3990 | . . . . . . 7 | |
8 | 7 | notbid 662 | . . . . . 6 |
9 | 3, 8 | syl5ibcom 154 | . . . . 5 |
10 | 6, 9 | jcad 305 | . . . 4 |
11 | ioran 747 | . . . 4 | |
12 | 10, 11 | syl6ibr 161 | . . 3 |
13 | 12 | adantr 274 | . 2 |
14 | sotritric.tri | . . 3 | |
15 | 3orrot 979 | . . . . . . 7 | |
16 | 3orcomb 982 | . . . . . . 7 | |
17 | 3orass 976 | . . . . . . 7 | |
18 | 15, 16, 17 | 3bitri 205 | . . . . . 6 |
19 | 18 | biimpi 119 | . . . . 5 |
20 | 19 | orcomd 724 | . . . 4 |
21 | 20 | ord 719 | . . 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 703 w3o 972 wceq 1348 wcel 2141 class class class wbr 3987 wor 4278 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-po 4279 df-iso 4280 |
This theorem is referenced by: distrlem4prl 7533 distrlem4pru 7534 |
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