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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 3777 | . . . . . 6 |
3 | 2 | notbid 592 | . . . . 5 |
4 | breq1 3767 | . . . . . . 7 | |
5 | 4 | anbi1d 438 | . . . . . 6 |
6 | breq1 3767 | . . . . . 6 | |
7 | 5, 6 | imbi12d 223 | . . . . 5 |
8 | 3, 7 | anbi12d 442 | . . . 4 |
9 | 8 | imbi2d 219 | . . 3 |
10 | breq2 3768 | . . . . . . 7 | |
11 | breq1 3767 | . . . . . . 7 | |
12 | 10, 11 | anbi12d 442 | . . . . . 6 |
13 | 12 | imbi1d 220 | . . . . 5 |
14 | 13 | anbi2d 437 | . . . 4 |
15 | 14 | imbi2d 219 | . . 3 |
16 | breq2 3768 | . . . . . . 7 | |
17 | 16 | anbi2d 437 | . . . . . 6 |
18 | breq2 3768 | . . . . . 6 | |
19 | 17, 18 | imbi12d 223 | . . . . 5 |
20 | 19 | anbi2d 437 | . . . 4 |
21 | 20 | imbi2d 219 | . . 3 |
22 | df-po 4033 | . . . . . . . 8 | |
23 | r3al 2366 | . . . . . . . 8 | |
24 | 22, 23 | bitri 173 | . . . . . . 7 |
25 | 24 | biimpi 113 | . . . . . 6 |
26 | 25 | 19.21bbi 1451 | . . . . 5 |
27 | 26 | 19.21bi 1450 | . . . 4 |
28 | 27 | com12 27 | . . 3 |
29 | 9, 15, 21, 28 | vtocl3ga 2623 | . 2 |
30 | 29 | com12 27 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 w3a 885 wal 1241 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 wpo 4031 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 |
This theorem is referenced by: poirr 4044 potr 4045 |
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