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Mirrors > Home > ILE Home > Th. List > sowlin | Unicode version |
Description: A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.) |
Ref | Expression |
---|---|
sowlin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3767 | . . . . 5 | |
2 | breq1 3767 | . . . . . 6 | |
3 | 2 | orbi1d 705 | . . . . 5 |
4 | 1, 3 | imbi12d 223 | . . . 4 |
5 | 4 | imbi2d 219 | . . 3 |
6 | breq2 3768 | . . . . 5 | |
7 | breq2 3768 | . . . . . 6 | |
8 | 7 | orbi2d 704 | . . . . 5 |
9 | 6, 8 | imbi12d 223 | . . . 4 |
10 | 9 | imbi2d 219 | . . 3 |
11 | breq2 3768 | . . . . . 6 | |
12 | breq1 3767 | . . . . . 6 | |
13 | 11, 12 | orbi12d 707 | . . . . 5 |
14 | 13 | imbi2d 219 | . . . 4 |
15 | 14 | imbi2d 219 | . . 3 |
16 | df-iso 4034 | . . . . 5 | |
17 | 3anass 889 | . . . . . . 7 | |
18 | rsp 2369 | . . . . . . . . 9 | |
19 | rsp2 2371 | . . . . . . . . 9 | |
20 | 18, 19 | syl6 29 | . . . . . . . 8 |
21 | 20 | impd 242 | . . . . . . 7 |
22 | 17, 21 | syl5bi 141 | . . . . . 6 |
23 | 22 | adantl 262 | . . . . 5 |
24 | 16, 23 | sylbi 114 | . . . 4 |
25 | 24 | com12 27 | . . 3 |
26 | 5, 10, 15, 25 | vtocl3ga 2623 | . 2 |
27 | 26 | impcom 116 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 w3a 885 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 wpo 4031 wor 4032 |
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-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-iso 4034 |
This theorem is referenced by: sotri2 4722 sotri3 4723 addextpr 6719 cauappcvgprlemloc 6750 caucvgprlemloc 6773 caucvgprprlemloc 6801 caucvgprprlemaddq 6806 ltsosr 6849 axpre-ltwlin 6957 xrlelttr 8722 xrltletr 8723 xrletr 8724 |
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