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Mirrors > Home > ILE Home > Th. List > po2nr | Unicode version |
Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
po2nr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poirr 4199 | . . 3 | |
2 | 1 | adantrr 470 | . 2 |
3 | potr 4200 | . . . . . 6 | |
4 | 3 | 3exp2 1188 | . . . . 5 |
5 | 4 | com34 83 | . . . 4 |
6 | 5 | pm2.43d 50 | . . 3 |
7 | 6 | imp32 255 | . 2 |
8 | 2, 7 | mtod 637 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wcel 1465 class class class wbr 3899 wpo 4186 |
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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-po 4188 |
This theorem is referenced by: po3nr 4202 so2nr 4213 tridc 6761 |
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