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 3098 | . . . . . . 7 | |
5 | 3, 4 | eqsstri 3029 | . . . . . 6 |
6 | 5 | ssbri 3827 | . . . . 5 |
7 | df-br 3786 | . . . . . 6 | |
8 | opelxp1 4395 | . . . . . 6 | |
9 | 7, 8 | sylbi 119 | . . . . 5 |
10 | 2, 6, 9 | 3syl 17 | . . . 4 |
11 | swoord.5 | . . . 4 | |
12 | swoord.4 | . . . 4 | |
13 | swoer.3 | . . . . 5 | |
14 | 13 | swopolem 4060 | . . . 4 |
15 | 1, 10, 11, 12, 14 | syl13anc 1171 | . . 3 |
16 | 3 | brdifun 6156 | . . . . . . 7 |
17 | 10, 12, 16 | syl2anc 403 | . . . . . 6 |
18 | 2, 17 | mpbid 145 | . . . . 5 |
19 | orc 665 | . . . . 5 | |
20 | 18, 19 | nsyl 590 | . . . 4 |
21 | biorf 695 | . . . 4 | |
22 | 20, 21 | syl 14 | . . 3 |
23 | 15, 22 | sylibrd 167 | . 2 |
24 | 13 | swopolem 4060 | . . . 4 |
25 | 1, 12, 11, 10, 24 | syl13anc 1171 | . . 3 |
26 | olc 664 | . . . . 5 | |
27 | 18, 26 | nsyl 590 | . . . 4 |
28 | biorf 695 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | 25, 29 | sylibrd 167 | . 2 |
31 | 23, 30 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 cdif 2970 cun 2971 cop 3401 class class class wbr 3785 cxp 4361 ccnv 4362 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |