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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 316 |
. . . 4
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3 | swopo.1 |
. . . . 5
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4 | 3 | ralrimivva 2455 |
. . . 4
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5 | breq1 3848 |
. . . . . 6
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6 | breq2 3849 |
. . . . . . 7
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7 | 6 | notbid 627 |
. . . . . 6
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8 | 5, 7 | imbi12d 232 |
. . . . 5
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9 | breq2 3849 |
. . . . . 6
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10 | breq1 3848 |
. . . . . . 7
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11 | 10 | notbid 627 |
. . . . . 6
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12 | 9, 11 | imbi12d 232 |
. . . . 5
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13 | 8, 12 | rspc2va 2735 |
. . . 4
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14 | 2, 4, 13 | syl2anr 284 |
. . 3
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15 | 14 | pm2.01d 583 |
. 2
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16 | 3 | 3adantr1 1102 |
. . 3
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17 | swopo.2 |
. . . . . . 7
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18 | 17 | imp 122 |
. . . . . 6
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19 | 18 | orcomd 683 |
. . . . 5
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20 | 19 | ord 678 |
. . . 4
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21 | 20 | expimpd 355 |
. . 3
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22 | 16, 21 | sylan2d 288 |
. 2
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23 | 15, 22 | ispod 4131 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-po 4123 |
This theorem is referenced by: swoer 6318 |
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