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Mirrors > Home > ILE Home > Th. List > asymref | Unicode version |
Description: Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 4866. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
asymref |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3786 | . . . . . . . . . . 11 | |
2 | vex 2604 | . . . . . . . . . . . 12 | |
3 | vex 2604 | . . . . . . . . . . . 12 | |
4 | 2, 3 | opeluu 4200 | . . . . . . . . . . 11 |
5 | 1, 4 | sylbi 119 | . . . . . . . . . 10 |
6 | 5 | simpld 110 | . . . . . . . . 9 |
7 | 6 | adantr 270 | . . . . . . . 8 |
8 | 7 | pm4.71ri 384 | . . . . . . 7 |
9 | 8 | bibi1i 226 | . . . . . 6 |
10 | elin 3155 | . . . . . . . 8 | |
11 | 2, 3 | brcnv 4536 | . . . . . . . . . 10 |
12 | df-br 3786 | . . . . . . . . . 10 | |
13 | 11, 12 | bitr3i 184 | . . . . . . . . 9 |
14 | 1, 13 | anbi12i 447 | . . . . . . . 8 |
15 | 10, 14 | bitr4i 185 | . . . . . . 7 |
16 | 3 | opelres 4635 | . . . . . . . 8 |
17 | df-br 3786 | . . . . . . . . . 10 | |
18 | 3 | ideq 4506 | . . . . . . . . . 10 |
19 | 17, 18 | bitr3i 184 | . . . . . . . . 9 |
20 | 19 | anbi2ci 446 | . . . . . . . 8 |
21 | 16, 20 | bitri 182 | . . . . . . 7 |
22 | 15, 21 | bibi12i 227 | . . . . . 6 |
23 | pm5.32 440 | . . . . . 6 | |
24 | 9, 22, 23 | 3bitr4i 210 | . . . . 5 |
25 | 24 | albii 1399 | . . . 4 |
26 | 19.21v 1794 | . . . 4 | |
27 | 25, 26 | bitri 182 | . . 3 |
28 | 27 | albii 1399 | . 2 |
29 | relcnv 4723 | . . . 4 | |
30 | relin2 4474 | . . . 4 | |
31 | 29, 30 | ax-mp 7 | . . 3 |
32 | relres 4657 | . . 3 | |
33 | eqrel 4447 | . . 3 | |
34 | 31, 32, 33 | mp2an 416 | . 2 |
35 | df-ral 2353 | . 2 | |
36 | 28, 34, 35 | 3bitr4i 210 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wceq 1284 wcel 1433 wral 2348 cin 2972 cop 3401 cuni 3601 class class class wbr 3785 cid 4043 ccnv 4362 cres 4365 wrel 4368 |
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-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-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-res 4375 |
This theorem is referenced by: (None) |
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