Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4846. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
asymref |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . . . . . . . . . . 11 | |
2 | vex 2560 | . . . . . . . . . . . 12 | |
3 | vex 2560 | . . . . . . . . . . . 12 | |
4 | 2, 3 | opeluu 4182 | . . . . . . . . . . 11 |
5 | 1, 4 | sylbi 114 | . . . . . . . . . 10 |
6 | 5 | simpld 105 | . . . . . . . . 9 |
7 | 6 | adantr 261 | . . . . . . . 8 |
8 | 7 | pm4.71ri 372 | . . . . . . 7 |
9 | 8 | bibi1i 217 | . . . . . 6 |
10 | elin 3126 | . . . . . . . 8 | |
11 | 2, 3 | brcnv 4518 | . . . . . . . . . 10 |
12 | df-br 3765 | . . . . . . . . . 10 | |
13 | 11, 12 | bitr3i 175 | . . . . . . . . 9 |
14 | 1, 13 | anbi12i 433 | . . . . . . . 8 |
15 | 10, 14 | bitr4i 176 | . . . . . . 7 |
16 | 3 | opelres 4617 | . . . . . . . 8 |
17 | df-br 3765 | . . . . . . . . . 10 | |
18 | 3 | ideq 4488 | . . . . . . . . . 10 |
19 | 17, 18 | bitr3i 175 | . . . . . . . . 9 |
20 | 19 | anbi2ci 432 | . . . . . . . 8 |
21 | 16, 20 | bitri 173 | . . . . . . 7 |
22 | 15, 21 | bibi12i 218 | . . . . . 6 |
23 | pm5.32 426 | . . . . . 6 | |
24 | 9, 22, 23 | 3bitr4i 201 | . . . . 5 |
25 | 24 | albii 1359 | . . . 4 |
26 | 19.21v 1753 | . . . 4 | |
27 | 25, 26 | bitri 173 | . . 3 |
28 | 27 | albii 1359 | . 2 |
29 | relcnv 4703 | . . . 4 | |
30 | relin2 4456 | . . . 4 | |
31 | 29, 30 | ax-mp 7 | . . 3 |
32 | relres 4639 | . . 3 | |
33 | eqrel 4429 | . . 3 | |
34 | 31, 32, 33 | mp2an 402 | . 2 |
35 | df-ral 2311 | . 2 | |
36 | 28, 34, 35 | 3bitr4i 201 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 wral 2306 cin 2916 cop 3378 cuni 3580 class class class wbr 3764 cid 4025 ccnv 4344 cres 4347 wrel 4350 |
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-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-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-res 4357 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |