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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 321 | . . . 4 |
3 | swopo.1 | . . . . 5 | |
4 | 3 | ralrimivva 2514 | . . . 4 |
5 | breq1 3932 | . . . . . 6 | |
6 | breq2 3933 | . . . . . . 7 | |
7 | 6 | notbid 656 | . . . . . 6 |
8 | 5, 7 | imbi12d 233 | . . . . 5 |
9 | breq2 3933 | . . . . . 6 | |
10 | breq1 3932 | . . . . . . 7 | |
11 | 10 | notbid 656 | . . . . . 6 |
12 | 9, 11 | imbi12d 233 | . . . . 5 |
13 | 8, 12 | rspc2va 2803 | . . . 4 |
14 | 2, 4, 13 | syl2anr 288 | . . 3 |
15 | 14 | pm2.01d 607 | . 2 |
16 | 3 | 3adantr1 1140 | . . 3 |
17 | swopo.2 | . . . . . . 7 | |
18 | 17 | imp 123 | . . . . . 6 |
19 | 18 | orcomd 718 | . . . . 5 |
20 | 19 | ord 713 | . . . 4 |
21 | 20 | expimpd 360 | . . 3 |
22 | 16, 21 | sylan2d 292 | . 2 |
23 | 15, 22 | ispod 4226 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 w3a 962 wcel 1480 wral 2416 class class class wbr 3929 wpo 4216 |
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 603 ax-in2 604 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-po 4218 |
This theorem is referenced by: swoer 6457 |
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