Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3902 | . . . . 5 | |
2 | breq1 3902 | . . . . . 6 | |
3 | 2 | orbi1d 765 | . . . . 5 |
4 | 1, 3 | imbi12d 233 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | breq2 3903 | . . . . 5 | |
7 | breq2 3903 | . . . . . 6 | |
8 | 7 | orbi2d 764 | . . . . 5 |
9 | 6, 8 | imbi12d 233 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | breq2 3903 | . . . . . 6 | |
12 | breq1 3902 | . . . . . 6 | |
13 | 11, 12 | orbi12d 767 | . . . . 5 |
14 | 13 | imbi2d 229 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | df-iso 4189 | . . . . 5 | |
17 | 3anass 951 | . . . . . . 7 | |
18 | rsp 2457 | . . . . . . . . 9 | |
19 | rsp2 2459 | . . . . . . . . 9 | |
20 | 18, 19 | syl6 33 | . . . . . . . 8 |
21 | 20 | impd 252 | . . . . . . 7 |
22 | 17, 21 | syl5bi 151 | . . . . . 6 |
23 | 22 | adantl 275 | . . . . 5 |
24 | 16, 23 | sylbi 120 | . . . 4 |
25 | 24 | com12 30 | . . 3 |
26 | 5, 10, 15, 25 | vtocl3ga 2730 | . 2 |
27 | 26 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 w3a 947 wceq 1316 wcel 1465 wral 2393 class class class wbr 3899 wpo 4186 wor 4187 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-iso 4189 |
This theorem is referenced by: sotri2 4906 sotri3 4907 suplub2ti 6856 addextpr 7397 cauappcvgprlemloc 7428 caucvgprlemloc 7451 caucvgprprlemloc 7479 caucvgprprlemaddq 7484 ltsosr 7540 suplocsrlem 7584 axpre-ltwlin 7659 xrlelttr 9557 xrltletr 9558 xrletr 9559 xrmaxiflemlub 10985 |
Copyright terms: Public domain | W3C validator |