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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 3942 | . . . . . 6 |
3 | 2 | notbid 656 | . . . . 5 |
4 | breq1 3932 | . . . . . . 7 | |
5 | 4 | anbi1d 460 | . . . . . 6 |
6 | breq1 3932 | . . . . . 6 | |
7 | 5, 6 | imbi12d 233 | . . . . 5 |
8 | 3, 7 | anbi12d 464 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | breq2 3933 | . . . . . . 7 | |
11 | breq1 3932 | . . . . . . 7 | |
12 | 10, 11 | anbi12d 464 | . . . . . 6 |
13 | 12 | imbi1d 230 | . . . . 5 |
14 | 13 | anbi2d 459 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | breq2 3933 | . . . . . . 7 | |
17 | 16 | anbi2d 459 | . . . . . 6 |
18 | breq2 3933 | . . . . . 6 | |
19 | 17, 18 | imbi12d 233 | . . . . 5 |
20 | 19 | anbi2d 459 | . . . 4 |
21 | 20 | imbi2d 229 | . . 3 |
22 | df-po 4218 | . . . . . . . 8 | |
23 | r3al 2477 | . . . . . . . 8 | |
24 | 22, 23 | bitri 183 | . . . . . . 7 |
25 | 24 | biimpi 119 | . . . . . 6 |
26 | 25 | 19.21bbi 1538 | . . . . 5 |
27 | 26 | 19.21bi 1537 | . . . 4 |
28 | 27 | com12 30 | . . 3 |
29 | 9, 15, 21, 28 | vtocl3ga 2756 | . 2 |
30 | 29 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2416 class class class wbr 3929 wpo 4216 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-po 4218 |
This theorem is referenced by: poirr 4229 potr 4230 |
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