Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4657 | . . . 4 | |
2 | relin2 4474 | . . . 4 | |
3 | 1, 2 | mp1i 10 | . . 3 |
4 | df-br 3786 | . . . . 5 | |
5 | brin 3832 | . . . . 5 | |
6 | 4, 5 | bitr3i 184 | . . . 4 |
7 | vex 2604 | . . . . . . . . 9 | |
8 | 7 | brres 4636 | . . . . . . . 8 |
9 | poirr 4062 | . . . . . . . . . . 11 | |
10 | 7 | ideq 4506 | . . . . . . . . . . . . 13 |
11 | breq2 3789 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | sylbi 119 | . . . . . . . . . . . 12 |
13 | 12 | notbid 624 | . . . . . . . . . . 11 |
14 | 9, 13 | syl5ibcom 153 | . . . . . . . . . 10 |
15 | 14 | expimpd 355 | . . . . . . . . 9 |
16 | 15 | ancomsd 265 | . . . . . . . 8 |
17 | 8, 16 | syl5bi 150 | . . . . . . 7 |
18 | 17 | con2d 586 | . . . . . 6 |
19 | imnan 656 | . . . . . 6 | |
20 | 18, 19 | sylib 120 | . . . . 5 |
21 | 20 | pm2.21d 581 | . . . 4 |
22 | 6, 21 | syl5bi 150 | . . 3 |
23 | 3, 22 | relssdv 4450 | . 2 |
24 | ss0 3284 | . 2 | |
25 | 23, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cin 2972 wss 2973 c0 3251 cop 3401 class class class wbr 3785 cid 4043 wpo 4049 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-in1 576 ax-in2 577 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-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-xp 4369 df-rel 4370 df-res 4375 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |