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Mirrors > Home > ILE Home > Th. List > swopo | Unicode version |
Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
swopo.1 |
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swopo.2 |
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Ref | Expression |
---|---|
swopo |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . 5
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2 | 1 | ancli 306 |
. . . 4
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3 | swopo.1 |
. . . . 5
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4 | 3 | ralrimivva 2395 |
. . . 4
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5 | breq1 3758 |
. . . . . 6
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6 | breq2 3759 |
. . . . . . 7
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7 | 6 | notbid 591 |
. . . . . 6
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8 | 5, 7 | imbi12d 223 |
. . . . 5
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9 | breq2 3759 |
. . . . . 6
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10 | breq1 3758 |
. . . . . . 7
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11 | 10 | notbid 591 |
. . . . . 6
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12 | 9, 11 | imbi12d 223 |
. . . . 5
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13 | 8, 12 | rspc2va 2657 |
. . . 4
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14 | 2, 4, 13 | syl2anr 274 |
. . 3
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15 | 14 | pm2.01d 548 |
. 2
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16 | 3 | 3adantr1 1062 |
. . 3
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17 | swopo.2 |
. . . . . . 7
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18 | 17 | imp 115 |
. . . . . 6
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19 | 18 | orcomd 647 |
. . . . 5
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20 | 19 | ord 642 |
. . . 4
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21 | 20 | expimpd 345 |
. . 3
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22 | 16, 21 | sylan2d 278 |
. 2
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23 | 15, 22 | ispod 4032 |
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-in1 544 ax-in2 545 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-po 4024 |
This theorem is referenced by: swoer 6070 |
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