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Mirrors > Home > ILE Home > Th. List > swopo | Unicode version |
Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
swopo.1 | |
swopo.2 |
Ref | Expression |
---|---|
swopo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . 5 | |
2 | 1 | ancli 316 | . . . 4 |
3 | swopo.1 | . . . . 5 | |
4 | 3 | ralrimivva 2443 | . . . 4 |
5 | breq1 3788 | . . . . . 6 | |
6 | breq2 3789 | . . . . . . 7 | |
7 | 6 | notbid 624 | . . . . . 6 |
8 | 5, 7 | imbi12d 232 | . . . . 5 |
9 | breq2 3789 | . . . . . 6 | |
10 | breq1 3788 | . . . . . . 7 | |
11 | 10 | notbid 624 | . . . . . 6 |
12 | 9, 11 | imbi12d 232 | . . . . 5 |
13 | 8, 12 | rspc2va 2714 | . . . 4 |
14 | 2, 4, 13 | syl2anr 284 | . . 3 |
15 | 14 | pm2.01d 580 | . 2 |
16 | 3 | 3adantr1 1097 | . . 3 |
17 | swopo.2 | . . . . . . 7 | |
18 | 17 | imp 122 | . . . . . 6 |
19 | 18 | orcomd 680 | . . . . 5 |
20 | 19 | ord 675 | . . . 4 |
21 | 20 | expimpd 355 | . . 3 |
22 | 16, 21 | sylan2d 288 | . 2 |
23 | 15, 22 | ispod 4059 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 w3a 919 wcel 1433 wral 2348 class class class wbr 3785 wpo 4049 |
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-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 |
This theorem is referenced by: swoer 6157 |
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