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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 3798 | . . . . . 6 |
3 | 2 | notbid 624 | . . . . 5 |
4 | breq1 3788 | . . . . . . 7 | |
5 | 4 | anbi1d 452 | . . . . . 6 |
6 | breq1 3788 | . . . . . 6 | |
7 | 5, 6 | imbi12d 232 | . . . . 5 |
8 | 3, 7 | anbi12d 456 | . . . 4 |
9 | 8 | imbi2d 228 | . . 3 |
10 | breq2 3789 | . . . . . . 7 | |
11 | breq1 3788 | . . . . . . 7 | |
12 | 10, 11 | anbi12d 456 | . . . . . 6 |
13 | 12 | imbi1d 229 | . . . . 5 |
14 | 13 | anbi2d 451 | . . . 4 |
15 | 14 | imbi2d 228 | . . 3 |
16 | breq2 3789 | . . . . . . 7 | |
17 | 16 | anbi2d 451 | . . . . . 6 |
18 | breq2 3789 | . . . . . 6 | |
19 | 17, 18 | imbi12d 232 | . . . . 5 |
20 | 19 | anbi2d 451 | . . . 4 |
21 | 20 | imbi2d 228 | . . 3 |
22 | df-po 4051 | . . . . . . . 8 | |
23 | r3al 2408 | . . . . . . . 8 | |
24 | 22, 23 | bitri 182 | . . . . . . 7 |
25 | 24 | biimpi 118 | . . . . . 6 |
26 | 25 | 19.21bbi 1491 | . . . . 5 |
27 | 26 | 19.21bi 1490 | . . . 4 |
28 | 27 | com12 30 | . . 3 |
29 | 9, 15, 21, 28 | vtocl3ga 2668 | . 2 |
30 | 29 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 w3a 919 wal 1282 wceq 1284 wcel 1433 wral 2348 class class class wbr 3785 wpo 4049 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-po 4051 |
This theorem is referenced by: poirr 4062 potr 4063 |
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