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Mirrors > Home > ILE Home > Th. List > pocl | Unicode version |
Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
pocl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . . . 7
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2 | 1, 1 | breq12d 3908 |
. . . . . 6
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3 | 2 | notbid 639 |
. . . . 5
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4 | breq1 3898 |
. . . . . . 7
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5 | 4 | anbi1d 458 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | breq1 3898 |
. . . . . 6
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7 | 5, 6 | imbi12d 233 |
. . . . 5
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8 | 3, 7 | anbi12d 462 |
. . . 4
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9 | 8 | imbi2d 229 |
. . 3
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10 | breq2 3899 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | breq1 3898 |
. . . . . . 7
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12 | 10, 11 | anbi12d 462 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 12 | imbi1d 230 |
. . . . 5
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14 | 13 | anbi2d 457 |
. . . 4
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15 | 14 | imbi2d 229 |
. . 3
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16 | breq2 3899 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | anbi2d 457 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | breq2 3899 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | imbi12d 233 |
. . . . 5
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20 | 19 | anbi2d 457 |
. . . 4
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21 | 20 | imbi2d 229 |
. . 3
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22 | df-po 4178 |
. . . . . . . 8
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23 | r3al 2451 |
. . . . . . . 8
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24 | 22, 23 | bitri 183 |
. . . . . . 7
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25 | 24 | biimpi 119 |
. . . . . 6
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26 | 25 | 19.21bbi 1521 |
. . . . 5
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27 | 26 | 19.21bi 1520 |
. . . 4
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28 | 27 | com12 30 |
. . 3
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29 | 9, 15, 21, 28 | vtocl3ga 2727 |
. 2
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30 | 29 | com12 30 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-po 4178 |
This theorem is referenced by: poirr 4189 potr 4190 |
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