Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > swoer | Unicode version |
Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
swoer.1 | |
swoer.2 | |
swoer.3 |
Ref | Expression |
---|---|
swoer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | swoer.1 | . . . . 5 | |
2 | difss 3172 | . . . . 5 | |
3 | 1, 2 | eqsstri 3099 | . . . 4 |
4 | relxp 4618 | . . . 4 | |
5 | relss 4596 | . . . 4 | |
6 | 3, 4, 5 | mp2 16 | . . 3 |
7 | 6 | a1i 9 | . 2 |
8 | simpr 109 | . . 3 | |
9 | orcom 702 | . . . . . 6 | |
10 | 9 | a1i 9 | . . . . 5 |
11 | 10 | notbid 641 | . . . 4 |
12 | 3 | ssbri 3942 | . . . . . . 7 |
13 | 12 | adantl 275 | . . . . . 6 |
14 | brxp 4540 | . . . . . 6 | |
15 | 13, 14 | sylib 121 | . . . . 5 |
16 | 1 | brdifun 6424 | . . . . 5 |
17 | 15, 16 | syl 14 | . . . 4 |
18 | 15 | simprd 113 | . . . . 5 |
19 | 15 | simpld 111 | . . . . 5 |
20 | 1 | brdifun 6424 | . . . . 5 |
21 | 18, 19, 20 | syl2anc 408 | . . . 4 |
22 | 11, 17, 21 | 3bitr4d 219 | . . 3 |
23 | 8, 22 | mpbid 146 | . 2 |
24 | simprl 505 | . . . . 5 | |
25 | 12 | ad2antrl 481 | . . . . . . 7 |
26 | 14 | simplbi 272 | . . . . . . 7 |
27 | 25, 26 | syl 14 | . . . . . 6 |
28 | 14 | simprbi 273 | . . . . . . 7 |
29 | 25, 28 | syl 14 | . . . . . 6 |
30 | 27, 29, 16 | syl2anc 408 | . . . . 5 |
31 | 24, 30 | mpbid 146 | . . . 4 |
32 | simprr 506 | . . . . 5 | |
33 | 3 | brel 4561 | . . . . . . . 8 |
34 | 33 | simprd 113 | . . . . . . 7 |
35 | 32, 34 | syl 14 | . . . . . 6 |
36 | 1 | brdifun 6424 | . . . . . 6 |
37 | 29, 35, 36 | syl2anc 408 | . . . . 5 |
38 | 32, 37 | mpbid 146 | . . . 4 |
39 | simpl 108 | . . . . . . 7 | |
40 | swoer.3 | . . . . . . . 8 | |
41 | 40 | swopolem 4197 | . . . . . . 7 |
42 | 39, 27, 35, 29, 41 | syl13anc 1203 | . . . . . 6 |
43 | 40 | swopolem 4197 | . . . . . . . 8 |
44 | 39, 35, 27, 29, 43 | syl13anc 1203 | . . . . . . 7 |
45 | orcom 702 | . . . . . . 7 | |
46 | 44, 45 | syl6ibr 161 | . . . . . 6 |
47 | 42, 46 | orim12d 760 | . . . . 5 |
48 | or4 745 | . . . . 5 | |
49 | 47, 48 | syl6ib 160 | . . . 4 |
50 | 31, 38, 49 | mtord 757 | . . 3 |
51 | 1 | brdifun 6424 | . . . 4 |
52 | 27, 35, 51 | syl2anc 408 | . . 3 |
53 | 50, 52 | mpbird 166 | . 2 |
54 | swoer.2 | . . . . . . 7 | |
55 | 54, 40 | swopo 4198 | . . . . . 6 |
56 | poirr 4199 | . . . . . 6 | |
57 | 55, 56 | sylan 281 | . . . . 5 |
58 | pm1.2 730 | . . . . 5 | |
59 | 57, 58 | nsyl 602 | . . . 4 |
60 | simpr 109 | . . . . 5 | |
61 | 1 | brdifun 6424 | . . . . 5 |
62 | 60, 60, 61 | syl2anc 408 | . . . 4 |
63 | 59, 62 | mpbird 166 | . . 3 |
64 | 3 | ssbri 3942 | . . . . 5 |
65 | brxp 4540 | . . . . . 6 | |
66 | 65 | simplbi 272 | . . . . 5 |
67 | 64, 66 | syl 14 | . . . 4 |
68 | 67 | adantl 275 | . . 3 |
69 | 63, 68 | impbida 570 | . 2 |
70 | 7, 23, 53, 69 | iserd 6423 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3a 947 wceq 1316 wcel 1465 cdif 3038 cun 3039 wss 3041 class class class wbr 3899 wpo 4186 cxp 4507 ccnv 4508 wrel 4514 wer 6394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-po 4188 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-er 6397 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |