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Mirrors > Home > MPE Home > Th. List > swopo | Structured version Visualization version 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 22 | . . . . 5 | |
2 | 1 | ancli 574 | . . . 4 |
3 | swopo.1 | . . . . 5 | |
4 | 3 | ralrimivva 2971 | . . . 4 |
5 | breq1 4656 | . . . . . 6 | |
6 | breq2 4657 | . . . . . . 7 | |
7 | 6 | notbid 308 | . . . . . 6 |
8 | 5, 7 | imbi12d 334 | . . . . 5 |
9 | breq2 4657 | . . . . . 6 | |
10 | breq1 4656 | . . . . . . 7 | |
11 | 10 | notbid 308 | . . . . . 6 |
12 | 9, 11 | imbi12d 334 | . . . . 5 |
13 | 8, 12 | rspc2va 3323 | . . . 4 |
14 | 2, 4, 13 | syl2anr 495 | . . 3 |
15 | 14 | pm2.01d 181 | . 2 |
16 | 3 | 3adantr1 1220 | . . 3 |
17 | swopo.2 | . . . . . . 7 | |
18 | 17 | imp 445 | . . . . . 6 |
19 | 18 | orcomd 403 | . . . . 5 |
20 | 19 | ord 392 | . . . 4 |
21 | 20 | expimpd 629 | . . 3 |
22 | 16, 21 | sylan2d 499 | . 2 |
23 | 15, 22 | ispod 5043 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wa 384 w3a 1037 wcel 1990 wral 2912 class class class wbr 4653 wpo 5033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-po 5035 |
This theorem is referenced by: swoer 7772 swoso 7775 |
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