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Mirrors > Home > MPE Home > Th. List > swoord1 | Structured version Visualization version 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 22 | . . . 4 | |
2 | swoord.6 | . . . . 5 | |
3 | swoer.1 | . . . . . . 7 | |
4 | difss 3737 | . . . . . . 7 | |
5 | 3, 4 | eqsstri 3635 | . . . . . 6 |
6 | 5 | ssbri 4697 | . . . . 5 |
7 | df-br 4654 | . . . . . 6 | |
8 | opelxp1 5150 | . . . . . 6 | |
9 | 7, 8 | sylbi 207 | . . . . 5 |
10 | 2, 6, 9 | 3syl 18 | . . . 4 |
11 | swoord.5 | . . . 4 | |
12 | swoord.4 | . . . 4 | |
13 | swoer.3 | . . . . 5 | |
14 | 13 | swopolem 5044 | . . . 4 |
15 | 1, 10, 11, 12, 14 | syl13anc 1328 | . . 3 |
16 | 3 | brdifun 7771 | . . . . . . 7 |
17 | 10, 12, 16 | syl2anc 693 | . . . . . 6 |
18 | 2, 17 | mpbid 222 | . . . . 5 |
19 | orc 400 | . . . . 5 | |
20 | 18, 19 | nsyl 135 | . . . 4 |
21 | biorf 420 | . . . 4 | |
22 | 20, 21 | syl 17 | . . 3 |
23 | 15, 22 | sylibrd 249 | . 2 |
24 | 13 | swopolem 5044 | . . . 4 |
25 | 1, 12, 11, 10, 24 | syl13anc 1328 | . . 3 |
26 | olc 399 | . . . . 5 | |
27 | 18, 26 | nsyl 135 | . . . 4 |
28 | biorf 420 | . . . 4 | |
29 | 27, 28 | syl 17 | . . 3 |
30 | 25, 29 | sylibrd 249 | . 2 |
31 | 23, 30 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 cdif 3571 cun 3572 cop 4183 class class class wbr 4653 cxp 5112 ccnv 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 |
This theorem is referenced by: (None) |
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