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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 3758 |
. . . . 5
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2 | breq1 3758 |
. . . . . 6
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3 | 2 | orbi1d 704 |
. . . . 5
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4 | 1, 3 | imbi12d 223 |
. . . 4
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5 | 4 | imbi2d 219 |
. . 3
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6 | breq2 3759 |
. . . . 5
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7 | breq2 3759 |
. . . . . 6
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8 | 7 | orbi2d 703 |
. . . . 5
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9 | 6, 8 | imbi12d 223 |
. . . 4
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10 | 9 | imbi2d 219 |
. . 3
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11 | breq2 3759 |
. . . . . 6
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12 | breq1 3758 |
. . . . . 6
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13 | 11, 12 | orbi12d 706 |
. . . . 5
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14 | 13 | imbi2d 219 |
. . . 4
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15 | 14 | imbi2d 219 |
. . 3
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16 | df-iso 4025 |
. . . . 5
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17 | 3anass 888 |
. . . . . . 7
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18 | rsp 2363 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | rsp2 2365 |
. . . . . . . . 9
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20 | 18, 19 | syl6 29 |
. . . . . . . 8
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21 | 20 | impd 242 |
. . . . . . 7
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22 | 17, 21 | syl5bi 141 |
. . . . . 6
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23 | 22 | adantl 262 |
. . . . 5
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24 | 16, 23 | sylbi 114 |
. . . 4
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25 | 24 | com12 27 |
. . 3
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26 | 5, 10, 15, 25 | vtocl3ga 2617 |
. 2
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27 | 26 | impcom 116 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-iso 4025 |
This theorem is referenced by: sotri2 4665 sotri3 4666 addextpr 6593 cauappcvgprlemloc 6624 caucvgprlemloc 6646 ltsosr 6692 axpre-ltwlin 6767 xrlelttr 8492 xrltletr 8493 xrletr 8494 |
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