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Mirrors > Home > MPE Home > Th. List > pocl | Structured version Visualization version 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 22 | . . . . . . 7 | |
2 | 1, 1 | breq12d 4666 | . . . . . 6 |
3 | 2 | notbid 308 | . . . . 5 |
4 | breq1 4656 | . . . . . . 7 | |
5 | 4 | anbi1d 741 | . . . . . 6 |
6 | breq1 4656 | . . . . . 6 | |
7 | 5, 6 | imbi12d 334 | . . . . 5 |
8 | 3, 7 | anbi12d 747 | . . . 4 |
9 | 8 | imbi2d 330 | . . 3 |
10 | breq2 4657 | . . . . . . 7 | |
11 | breq1 4656 | . . . . . . 7 | |
12 | 10, 11 | anbi12d 747 | . . . . . 6 |
13 | 12 | imbi1d 331 | . . . . 5 |
14 | 13 | anbi2d 740 | . . . 4 |
15 | 14 | imbi2d 330 | . . 3 |
16 | breq2 4657 | . . . . . . 7 | |
17 | 16 | anbi2d 740 | . . . . . 6 |
18 | breq2 4657 | . . . . . 6 | |
19 | 17, 18 | imbi12d 334 | . . . . 5 |
20 | 19 | anbi2d 740 | . . . 4 |
21 | 20 | imbi2d 330 | . . 3 |
22 | df-po 5035 | . . . . . . 7 | |
23 | r3al 2940 | . . . . . . 7 | |
24 | 22, 23 | sylbb 209 | . . . . . 6 |
25 | 24 | 19.21bbi 2060 | . . . . 5 |
26 | 25 | 19.21bi 2059 | . . . 4 |
27 | 26 | com12 32 | . . 3 |
28 | 9, 15, 21, 27 | vtocl3ga 3276 | . 2 |
29 | 28 | com12 32 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 w3a 1037 wal 1481 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653 wpo 5033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-po 5035 |
This theorem is referenced by: poirr 5046 potr 5047 |
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