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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 3243 | . . . . . . 7 | |
5 | 3, 4 | eqsstri 3169 | . . . . . 6 |
6 | 5 | ssbri 4020 | . . . . 5 |
7 | df-br 3977 | . . . . . 6 | |
8 | opelxp1 4632 | . . . . . 6 | |
9 | 7, 8 | sylbi 120 | . . . . 5 |
10 | 2, 6, 9 | 3syl 17 | . . . 4 |
11 | swoord.5 | . . . 4 | |
12 | swoord.4 | . . . 4 | |
13 | swoer.3 | . . . . 5 | |
14 | 13 | swopolem 4277 | . . . 4 |
15 | 1, 10, 11, 12, 14 | syl13anc 1229 | . . 3 |
16 | 3 | brdifun 6519 | . . . . . . 7 |
17 | 10, 12, 16 | syl2anc 409 | . . . . . 6 |
18 | 2, 17 | mpbid 146 | . . . . 5 |
19 | orc 702 | . . . . 5 | |
20 | 18, 19 | nsyl 618 | . . . 4 |
21 | biorf 734 | . . . 4 | |
22 | 20, 21 | syl 14 | . . 3 |
23 | 15, 22 | sylibrd 168 | . 2 |
24 | 13 | swopolem 4277 | . . . 4 |
25 | 1, 12, 11, 10, 24 | syl13anc 1229 | . . 3 |
26 | olc 701 | . . . . 5 | |
27 | 18, 26 | nsyl 618 | . . . 4 |
28 | biorf 734 | . . . 4 | |
29 | 27, 28 | syl 14 | . . 3 |
30 | 25, 29 | sylibrd 168 | . 2 |
31 | 23, 30 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wcel 2135 cdif 3108 cun 3109 cop 3573 class class class wbr 3976 cxp 4596 ccnv 4597 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 |
This theorem is referenced by: (None) |
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