Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > swoord1 | Unicode version |
Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.) |
Ref | Expression |
---|---|
swoer.1 | |
swoer.2 | |
swoer.3 | |
swoord.4 | |
swoord.5 | |
swoord.6 |
Ref | Expression |
---|---|
swoord1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 | |
2 | swoord.6 | . . . . 5 | |
3 | swoer.1 | . . . . . . 7 | |
4 | difss 3070 | . . . . . . 7 | |
5 | 3, 4 | eqsstri 2975 | . . . . . 6 |
6 | 5 | ssbri 3806 | . . . . 5 |
7 | df-br 3765 | . . . . . 6 | |
8 | opelxp1 4377 | . . . . . 6 | |
9 | 7, 8 | sylbi 114 | . . . . 5 |
10 | 2, 6, 9 | 3syl 17 | . . . 4 |
11 | swoord.5 | . . . 4 | |
12 | swoord.4 | . . . 4 | |
13 | swoer.3 | . . . . 5 | |
14 | 13 | swopolem 4042 | . . . 4 |
15 | 1, 10, 11, 12, 14 | syl13anc 1137 | . . 3 |
16 | 3 | brdifun 6133 | . . . . . . 7 |
17 | 10, 12, 16 | syl2anc 391 | . . . . . 6 |
18 | 2, 17 | mpbid 135 | . . . . 5 |
19 | orc 633 | . . . . 5 | |
20 | 18, 19 | nsyl 558 | . . . 4 |
21 | biorf 663 | . . . 4 | |
22 | 20, 21 | syl 14 | . . 3 |
23 | 15, 22 | sylibrd 158 | . 2 |
24 | 13 | swopolem 4042 | . . . 4 |
25 | 1, 12, 11, 10, 24 | syl13anc 1137 | . . 3 |
26 | olc 632 | . . . . 5 | |
27 | 18, 26 | nsyl 558 | . . . 4 |
28 | biorf 663 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | 25, 29 | sylibrd 158 | . 2 |
31 | 23, 30 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wcel 1393 cdif 2914 cun 2915 cop 3378 class class class wbr 3764 cxp 4343 ccnv 4344 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |