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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 306 | . . . 4 |
3 | swopo.1 | . . . . 5 | |
4 | 3 | ralrimivva 2401 | . . . 4 |
5 | breq1 3767 | . . . . . 6 | |
6 | breq2 3768 | . . . . . . 7 | |
7 | 6 | notbid 592 | . . . . . 6 |
8 | 5, 7 | imbi12d 223 | . . . . 5 |
9 | breq2 3768 | . . . . . 6 | |
10 | breq1 3767 | . . . . . . 7 | |
11 | 10 | notbid 592 | . . . . . 6 |
12 | 9, 11 | imbi12d 223 | . . . . 5 |
13 | 8, 12 | rspc2va 2663 | . . . 4 |
14 | 2, 4, 13 | syl2anr 274 | . . 3 |
15 | 14 | pm2.01d 548 | . 2 |
16 | 3 | 3adantr1 1063 | . . 3 |
17 | swopo.2 | . . . . . . 7 | |
18 | 17 | imp 115 | . . . . . 6 |
19 | 18 | orcomd 648 | . . . . 5 |
20 | 19 | ord 643 | . . . 4 |
21 | 20 | expimpd 345 | . . 3 |
22 | 16, 21 | sylan2d 278 | . 2 |
23 | 15, 22 | ispod 4041 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 w3a 885 wcel 1393 wral 2306 class class class wbr 3764 wpo 4031 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 |
This theorem is referenced by: swoer 6134 |
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