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Mirrors > Home > MPE Home > Th. List > swopo | Structured version Visualization version 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 22 |
. . . . 5
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2 | 1 | ancli 554 |
. . . 4
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3 | swopo.1 |
. . . . 5
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4 | 3 | ralrimivva 2808 |
. . . 4
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5 | breq1 4404 |
. . . . . 6
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6 | breq2 4405 |
. . . . . . 7
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7 | 6 | notbid 296 |
. . . . . 6
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8 | 5, 7 | imbi12d 322 |
. . . . 5
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9 | breq2 4405 |
. . . . . 6
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10 | breq1 4404 |
. . . . . . 7
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11 | 10 | notbid 296 |
. . . . . 6
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12 | 9, 11 | imbi12d 322 |
. . . . 5
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13 | 8, 12 | rspc2va 3159 |
. . . 4
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14 | 2, 4, 13 | syl2anr 481 |
. . 3
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15 | 14 | pm2.01d 173 |
. 2
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16 | 3 | 3adantr1 1166 |
. . 3
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17 | swopo.2 |
. . . . . . 7
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18 | 17 | imp 431 |
. . . . . 6
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19 | 18 | orcomd 390 |
. . . . 5
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20 | 19 | ord 379 |
. . . 4
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21 | 20 | expimpd 607 |
. . 3
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22 | 16, 21 | sylan2d 485 |
. 2
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23 | 15, 22 | ispod 4762 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ral 2741 df-rab 2745 df-v 3046 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-sn 3968 df-pr 3970 df-op 3974 df-br 4402 df-po 4754 |
This theorem is referenced by: swoer 7388 swoso 7391 |
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