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Mirrors > Home > ILE Home > Th. List > soirri | Unicode version |
Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
soi.1 | |
soi.2 |
Ref | Expression |
---|---|
soirri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 | |
2 | soi.1 | . . 3 | |
3 | soi.2 | . . . . 5 | |
4 | 3 | brel 4410 | . . . 4 |
5 | 4 | simpld 110 | . . 3 |
6 | sonr 4072 | . . 3 | |
7 | 2, 5, 6 | sylancr 405 | . 2 |
8 | 1, 7 | pm2.65i 600 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wcel 1433 wss 2973 class class class wbr 3785 wor 4050 cxp 4361 |
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-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-po 4051 df-iso 4052 df-xp 4369 |
This theorem is referenced by: son2lpi 4741 ltsonq 6588 genpdisj 6713 ltposr 6940 axpre-ltirr 7048 |
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