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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . . 7 | |
2 | 1, 1 | breq12d 3989 | . . . . . 6 |
3 | 2 | notbid 657 | . . . . 5 |
4 | breq1 3979 | . . . . . . 7 | |
5 | 4 | anbi1d 461 | . . . . . 6 |
6 | breq1 3979 | . . . . . 6 | |
7 | 5, 6 | imbi12d 233 | . . . . 5 |
8 | 3, 7 | anbi12d 465 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | breq2 3980 | . . . . . . 7 | |
11 | breq1 3979 | . . . . . . 7 | |
12 | 10, 11 | anbi12d 465 | . . . . . 6 |
13 | 12 | imbi1d 230 | . . . . 5 |
14 | 13 | anbi2d 460 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | breq2 3980 | . . . . . . 7 | |
17 | 16 | anbi2d 460 | . . . . . 6 |
18 | breq2 3980 | . . . . . 6 | |
19 | 17, 18 | imbi12d 233 | . . . . 5 |
20 | 19 | anbi2d 460 | . . . 4 |
21 | 20 | imbi2d 229 | . . 3 |
22 | df-po 4268 | . . . . . . . 8 | |
23 | r3al 2508 | . . . . . . . 8 | |
24 | 22, 23 | bitri 183 | . . . . . . 7 |
25 | 24 | biimpi 119 | . . . . . 6 |
26 | 25 | 19.21bbi 1546 | . . . . 5 |
27 | 26 | 19.21bi 1545 | . . . 4 |
28 | 27 | com12 30 | . . 3 |
29 | 9, 15, 21, 28 | vtocl3ga 2791 | . 2 |
30 | 29 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 967 wal 1340 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 wpo 4266 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-po 4268 |
This theorem is referenced by: poirr 4279 potr 4280 |
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