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Mirrors > Home > ILE Home > Th. List > po3nr | Unicode version |
Description: A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
po3nr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | po2nr 4037 |
. . 3
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2 | 1 | 3adantr2 1063 |
. 2
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3 | df-3an 886 |
. . 3
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4 | potr 4036 |
. . . 4
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5 | 4 | anim1d 319 |
. . 3
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6 | 3, 5 | syl5bi 141 |
. 2
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7 | 2, 6 | mtod 588 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-po 4024 |
This theorem is referenced by: so3nr 4050 |
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