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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 3932 | . . . . 5 | |
2 | breq1 3932 | . . . . . 6 | |
3 | 2 | orbi1d 780 | . . . . 5 |
4 | 1, 3 | imbi12d 233 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | breq2 3933 | . . . . 5 | |
7 | breq2 3933 | . . . . . 6 | |
8 | 7 | orbi2d 779 | . . . . 5 |
9 | 6, 8 | imbi12d 233 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | breq2 3933 | . . . . . 6 | |
12 | breq1 3932 | . . . . . 6 | |
13 | 11, 12 | orbi12d 782 | . . . . 5 |
14 | 13 | imbi2d 229 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | df-iso 4219 | . . . . 5 | |
17 | 3anass 966 | . . . . . . 7 | |
18 | rsp 2480 | . . . . . . . . 9 | |
19 | rsp2 2482 | . . . . . . . . 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 2756 | . 2 |
27 | 26 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 wral 2416 class class class wbr 3929 wpo 4216 wor 4217 |
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 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-iso 4219 |
This theorem is referenced by: sotri2 4936 sotri3 4937 suplub2ti 6888 addextpr 7429 cauappcvgprlemloc 7460 caucvgprlemloc 7483 caucvgprprlemloc 7511 caucvgprprlemaddq 7516 ltsosr 7572 suplocsrlem 7616 axpre-ltwlin 7691 xrlelttr 9589 xrltletr 9590 xrletr 9591 xrmaxiflemlub 11017 |
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