Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cvrco | Structured version Visualization version Unicode version |
Description: The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.) |
Ref | Expression |
---|---|
1cvrco.b | |
1cvrco.u | |
1cvrco.o | |
1cvrco.c | |
1cvrco.a |
Ref | Expression |
---|---|
1cvrco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlop 34649 | . . . . 5 | |
2 | 1 | adantr 481 | . . . 4 |
3 | simpr 477 | . . . 4 | |
4 | 1cvrco.b | . . . . . 6 | |
5 | 1cvrco.u | . . . . . 6 | |
6 | 4, 5 | op1cl 34472 | . . . . 5 |
7 | 2, 6 | syl 17 | . . . 4 |
8 | 1cvrco.o | . . . . 5 | |
9 | 1cvrco.c | . . . . 5 | |
10 | 4, 8, 9 | cvrcon3b 34564 | . . . 4 |
11 | 2, 3, 7, 10 | syl3anc 1326 | . . 3 |
12 | eqid 2622 | . . . . . 6 | |
13 | 12, 5, 8 | opoc1 34489 | . . . . 5 |
14 | 2, 13 | syl 17 | . . . 4 |
15 | 14 | breq1d 4663 | . . 3 |
16 | 4, 8 | opoccl 34481 | . . . . 5 |
17 | 1, 16 | sylan 488 | . . . 4 |
18 | 17 | biantrurd 529 | . . 3 |
19 | 11, 15, 18 | 3bitrd 294 | . 2 |
20 | 1cvrco.a | . . . 4 | |
21 | 4, 12, 9, 20 | isat 34573 | . . 3 |
22 | 21 | adantr 481 | . 2 |
23 | 19, 22 | bitr4d 271 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 cbs 15857 coc 15949 cp0 17037 cp1 17038 cops 34459 ccvr 34549 catm 34550 chlt 34637 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-p0 17039 df-p1 17040 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-hlat 34638 |
This theorem is referenced by: 1cvratex 34759 lhpoc 35300 |
Copyright terms: Public domain | W3C validator |