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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 3990 | . . . . 5 | |
2 | breq1 3990 | . . . . . 6 | |
3 | 2 | orbi1d 786 | . . . . 5 |
4 | 1, 3 | imbi12d 233 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | breq2 3991 | . . . . 5 | |
7 | breq2 3991 | . . . . . 6 | |
8 | 7 | orbi2d 785 | . . . . 5 |
9 | 6, 8 | imbi12d 233 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | breq2 3991 | . . . . . 6 | |
12 | breq1 3990 | . . . . . 6 | |
13 | 11, 12 | orbi12d 788 | . . . . 5 |
14 | 13 | imbi2d 229 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | df-iso 4280 | . . . . 5 | |
17 | 3anass 977 | . . . . . . 7 | |
18 | rsp 2517 | . . . . . . . . 9 | |
19 | rsp2 2520 | . . . . . . . . 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 2800 | . 2 |
27 | 26 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 w3a 973 wceq 1348 wcel 2141 wral 2448 class class class wbr 3987 wpo 4277 wor 4278 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-iso 4280 |
This theorem is referenced by: sotri2 5006 sotri3 5007 suplub2ti 6974 addextpr 7570 cauappcvgprlemloc 7601 caucvgprlemloc 7624 caucvgprprlemloc 7652 caucvgprprlemaddq 7657 ltsosr 7713 suplocsrlem 7757 axpre-ltwlin 7832 xrlelttr 9750 xrltletr 9751 xrletr 9752 xrmaxiflemlub 11198 |
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