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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 4013 |
. . . . . 6
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3 | 2 | notbid 667 |
. . . . 5
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4 | breq1 4003 |
. . . . . . 7
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5 | 4 | anbi1d 465 |
. . . . . 6
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6 | breq1 4003 |
. . . . . 6
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7 | 5, 6 | imbi12d 234 |
. . . . 5
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8 | 3, 7 | anbi12d 473 |
. . . 4
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9 | 8 | imbi2d 230 |
. . 3
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10 | breq2 4004 |
. . . . . . 7
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11 | breq1 4003 |
. . . . . . 7
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12 | 10, 11 | anbi12d 473 |
. . . . . 6
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13 | 12 | imbi1d 231 |
. . . . 5
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14 | 13 | anbi2d 464 |
. . . 4
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15 | 14 | imbi2d 230 |
. . 3
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16 | breq2 4004 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | anbi2d 464 |
. . . . . 6
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18 | breq2 4004 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | imbi12d 234 |
. . . . 5
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20 | 19 | anbi2d 464 |
. . . 4
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21 | 20 | imbi2d 230 |
. . 3
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22 | df-po 4292 |
. . . . . . . 8
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23 | r3al 2521 |
. . . . . . . 8
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24 | 22, 23 | bitri 184 |
. . . . . . 7
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25 | 24 | biimpi 120 |
. . . . . 6
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26 | 25 | 19.21bbi 1559 |
. . . . 5
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27 | 26 | 19.21bi 1558 |
. . . 4
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28 | 27 | com12 30 |
. . 3
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29 | 9, 15, 21, 28 | vtocl3ga 2807 |
. 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-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-po 4292 |
This theorem is referenced by: poirr 4303 potr 4304 |
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