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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 321 |
. . . 4
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3 | swopo.1 |
. . . . 5
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4 | 3 | ralrimivva 2517 |
. . . 4
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5 | breq1 3940 |
. . . . . 6
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6 | breq2 3941 |
. . . . . . 7
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7 | 6 | notbid 657 |
. . . . . 6
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8 | 5, 7 | imbi12d 233 |
. . . . 5
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9 | breq2 3941 |
. . . . . 6
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10 | breq1 3940 |
. . . . . . 7
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11 | 10 | notbid 657 |
. . . . . 6
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12 | 9, 11 | imbi12d 233 |
. . . . 5
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13 | 8, 12 | rspc2va 2807 |
. . . 4
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14 | 2, 4, 13 | syl2anr 288 |
. . 3
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15 | 14 | pm2.01d 608 |
. 2
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16 | 3 | 3adantr1 1141 |
. . 3
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17 | swopo.2 |
. . . . . . 7
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18 | 17 | imp 123 |
. . . . . 6
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19 | 18 | orcomd 719 |
. . . . 5
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20 | 19 | ord 714 |
. . . 4
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21 | 20 | expimpd 361 |
. . 3
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22 | 16, 21 | sylan2d 292 |
. 2
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23 | 15, 22 | ispod 4234 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-po 4226 |
This theorem is referenced by: swoer 6465 |
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