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Mirrors > Home > ILE Home > Th. List > poirr2 | Unicode version |
Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) |
Ref | Expression |
---|---|
poirr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4639 | . . . 4 | |
2 | relin2 4456 | . . . 4 | |
3 | 1, 2 | mp1i 10 | . . 3 |
4 | df-br 3765 | . . . . 5 | |
5 | brin 3811 | . . . . 5 | |
6 | 4, 5 | bitr3i 175 | . . . 4 |
7 | vex 2560 | . . . . . . . . 9 | |
8 | 7 | brres 4618 | . . . . . . . 8 |
9 | poirr 4044 | . . . . . . . . . . 11 | |
10 | 7 | ideq 4488 | . . . . . . . . . . . . 13 |
11 | breq2 3768 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | sylbi 114 | . . . . . . . . . . . 12 |
13 | 12 | notbid 592 | . . . . . . . . . . 11 |
14 | 9, 13 | syl5ibcom 144 | . . . . . . . . . 10 |
15 | 14 | expimpd 345 | . . . . . . . . 9 |
16 | 15 | ancomsd 256 | . . . . . . . 8 |
17 | 8, 16 | syl5bi 141 | . . . . . . 7 |
18 | 17 | con2d 554 | . . . . . 6 |
19 | imnan 624 | . . . . . 6 | |
20 | 18, 19 | sylib 127 | . . . . 5 |
21 | 20 | pm2.21d 549 | . . . 4 |
22 | 6, 21 | syl5bi 141 | . . 3 |
23 | 3, 22 | relssdv 4432 | . 2 |
24 | ss0 3257 | . 2 | |
25 | 23, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cin 2916 wss 2917 c0 3224 cop 3378 class class class wbr 3764 cid 4025 wpo 4031 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-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-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-po 4033 df-xp 4351 df-rel 4352 df-res 4357 |
This theorem is referenced by: (None) |
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