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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 4817 | . . . 4 | |
2 | relin2 4628 | . . . 4 | |
3 | 1, 2 | mp1i 10 | . . 3 |
4 | df-br 3900 | . . . . 5 | |
5 | brin 3950 | . . . . 5 | |
6 | 4, 5 | bitr3i 185 | . . . 4 |
7 | vex 2663 | . . . . . . . . 9 | |
8 | 7 | brres 4795 | . . . . . . . 8 |
9 | poirr 4199 | . . . . . . . . . . 11 | |
10 | 7 | ideq 4661 | . . . . . . . . . . . . 13 |
11 | breq2 3903 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | sylbi 120 | . . . . . . . . . . . 12 |
13 | 12 | notbid 641 | . . . . . . . . . . 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 598 | . . . . . 6 |
19 | imnan 664 | . . . . . 6 | |
20 | 18, 19 | sylib 121 | . . . . 5 |
21 | 20 | pm2.21d 593 | . . . 4 |
22 | 6, 21 | syl5bi 151 | . . 3 |
23 | 3, 22 | relssdv 4601 | . 2 |
24 | ss0 3373 | . 2 | |
25 | 23, 24 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cin 3040 wss 3041 c0 3333 cop 3500 class class class wbr 3899 cid 4180 wpo 4186 cres 4511 wrel 4514 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-xp 4515 df-rel 4516 df-res 4521 |
This theorem is referenced by: (None) |
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