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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4006 |
. . . . 5
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2 | breq1 4006 |
. . . . . 6
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3 | 2 | orbi1d 791 |
. . . . 5
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4 | 1, 3 | imbi12d 234 |
. . . 4
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5 | 4 | imbi2d 230 |
. . 3
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6 | breq2 4007 |
. . . . 5
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7 | breq2 4007 |
. . . . . 6
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8 | 7 | orbi2d 790 |
. . . . 5
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9 | 6, 8 | imbi12d 234 |
. . . 4
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10 | 9 | imbi2d 230 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | breq2 4007 |
. . . . . 6
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12 | breq1 4006 |
. . . . . 6
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13 | 11, 12 | orbi12d 793 |
. . . . 5
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14 | 13 | imbi2d 230 |
. . . 4
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15 | 14 | imbi2d 230 |
. . 3
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16 | df-iso 4297 |
. . . . 5
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17 | 3anass 982 |
. . . . . . 7
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18 | rsp 2524 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | rsp2 2527 |
. . . . . . . . 9
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20 | 18, 19 | syl6 33 |
. . . . . . . 8
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21 | 20 | impd 254 |
. . . . . . 7
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22 | 17, 21 | biimtrid 152 |
. . . . . 6
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23 | 22 | adantl 277 |
. . . . 5
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24 | 16, 23 | sylbi 121 |
. . . 4
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25 | 24 | com12 30 |
. . 3
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26 | 5, 10, 15, 25 | vtocl3ga 2807 |
. 2
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27 | 26 | impcom 125 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-iso 4297 |
This theorem is referenced by: sotri2 5026 sotri3 5027 suplub2ti 6999 addextpr 7619 cauappcvgprlemloc 7650 caucvgprlemloc 7673 caucvgprprlemloc 7701 caucvgprprlemaddq 7706 ltsosr 7762 suplocsrlem 7806 axpre-ltwlin 7881 xrlelttr 9804 xrltletr 9805 xrletr 9806 xrmaxiflemlub 11251 |
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