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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 3070 | . . . . 5 | |
3 | 1, 2 | eqsstri 2975 | . . . 4 |
4 | relxp 4447 | . . . 4 | |
5 | relss 4427 | . . . 4 | |
6 | 3, 4, 5 | mp2 16 | . . 3 |
7 | 6 | a1i 9 | . 2 |
8 | simpr 103 | . . 3 | |
9 | orcom 647 | . . . . . 6 | |
10 | 9 | a1i 9 | . . . . 5 |
11 | 10 | notbid 592 | . . . 4 |
12 | 3 | ssbri 3806 | . . . . . . 7 |
13 | 12 | adantl 262 | . . . . . 6 |
14 | brxp 4375 | . . . . . 6 | |
15 | 13, 14 | sylib 127 | . . . . 5 |
16 | 1 | brdifun 6133 | . . . . 5 |
17 | 15, 16 | syl 14 | . . . 4 |
18 | 15 | simprd 107 | . . . . 5 |
19 | 15 | simpld 105 | . . . . 5 |
20 | 1 | brdifun 6133 | . . . . 5 |
21 | 18, 19, 20 | syl2anc 391 | . . . 4 |
22 | 11, 17, 21 | 3bitr4d 209 | . . 3 |
23 | 8, 22 | mpbid 135 | . 2 |
24 | simprl 483 | . . . . 5 | |
25 | 12 | ad2antrl 459 | . . . . . . 7 |
26 | 14 | simplbi 259 | . . . . . . 7 |
27 | 25, 26 | syl 14 | . . . . . 6 |
28 | 14 | simprbi 260 | . . . . . . 7 |
29 | 25, 28 | syl 14 | . . . . . 6 |
30 | 27, 29, 16 | syl2anc 391 | . . . . 5 |
31 | 24, 30 | mpbid 135 | . . . 4 |
32 | simprr 484 | . . . . 5 | |
33 | 3 | brel 4392 | . . . . . . . 8 |
34 | 33 | simprd 107 | . . . . . . 7 |
35 | 32, 34 | syl 14 | . . . . . 6 |
36 | 1 | brdifun 6133 | . . . . . 6 |
37 | 29, 35, 36 | syl2anc 391 | . . . . 5 |
38 | 32, 37 | mpbid 135 | . . . 4 |
39 | simpl 102 | . . . . . . 7 | |
40 | swoer.3 | . . . . . . . 8 | |
41 | 40 | swopolem 4042 | . . . . . . 7 |
42 | 39, 27, 35, 29, 41 | syl13anc 1137 | . . . . . 6 |
43 | 40 | swopolem 4042 | . . . . . . . 8 |
44 | 39, 35, 27, 29, 43 | syl13anc 1137 | . . . . . . 7 |
45 | orcom 647 | . . . . . . 7 | |
46 | 44, 45 | syl6ibr 151 | . . . . . 6 |
47 | 42, 46 | orim12d 700 | . . . . 5 |
48 | or4 688 | . . . . 5 | |
49 | 47, 48 | syl6ib 150 | . . . 4 |
50 | 31, 38, 49 | mtord 697 | . . 3 |
51 | 1 | brdifun 6133 | . . . 4 |
52 | 27, 35, 51 | syl2anc 391 | . . 3 |
53 | 50, 52 | mpbird 156 | . 2 |
54 | swoer.2 | . . . . . . 7 | |
55 | 54, 40 | swopo 4043 | . . . . . 6 |
56 | poirr 4044 | . . . . . 6 | |
57 | 55, 56 | sylan 267 | . . . . 5 |
58 | pm1.2 673 | . . . . 5 | |
59 | 57, 58 | nsyl 558 | . . . 4 |
60 | simpr 103 | . . . . 5 | |
61 | 1 | brdifun 6133 | . . . . 5 |
62 | 60, 60, 61 | syl2anc 391 | . . . 4 |
63 | 59, 62 | mpbird 156 | . . 3 |
64 | 3 | ssbri 3806 | . . . . 5 |
65 | brxp 4375 | . . . . . 6 | |
66 | 65 | simplbi 259 | . . . . 5 |
67 | 64, 66 | syl 14 | . . . 4 |
68 | 67 | adantl 262 | . . 3 |
69 | 63, 68 | impbida 528 | . 2 |
70 | 7, 23, 53, 69 | iserd 6132 | 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 wss 2917 class class class wbr 3764 wpo 4031 cxp 4343 ccnv 4344 wrel 4350 wer 6103 |
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-po 4033 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-er 6106 |
This theorem is referenced by: (None) |
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