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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 4842 | . . . 4 | |
2 | relin2 4653 | . . . 4 | |
3 | 1, 2 | mp1i 10 | . . 3 |
4 | df-br 3925 | . . . . 5 | |
5 | brin 3975 | . . . . 5 | |
6 | 4, 5 | bitr3i 185 | . . . 4 |
7 | vex 2684 | . . . . . . . . 9 | |
8 | 7 | brres 4820 | . . . . . . . 8 |
9 | poirr 4224 | . . . . . . . . . . 11 | |
10 | 7 | ideq 4686 | . . . . . . . . . . . . 13 |
11 | breq2 3928 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | sylbi 120 | . . . . . . . . . . . 12 |
13 | 12 | notbid 656 | . . . . . . . . . . 11 |
14 | 9, 13 | syl5ibcom 154 | . . . . . . . . . 10 |
15 | 14 | expimpd 360 | . . . . . . . . 9 |
16 | 15 | ancomsd 267 | . . . . . . . 8 |
17 | 8, 16 | syl5bi 151 | . . . . . . 7 |
18 | 17 | con2d 613 | . . . . . 6 |
19 | imnan 679 | . . . . . 6 | |
20 | 18, 19 | sylib 121 | . . . . 5 |
21 | 20 | pm2.21d 608 | . . . 4 |
22 | 6, 21 | syl5bi 151 | . . 3 |
23 | 3, 22 | relssdv 4626 | . 2 |
24 | ss0 3398 | . 2 | |
25 | 23, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cin 3065 wss 3066 c0 3358 cop 3525 class class class wbr 3924 cid 4205 wpo 4211 cres 4536 wrel 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-xp 4540 df-rel 4541 df-res 4546 |
This theorem is referenced by: (None) |
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