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Theorem poirr 4035
Description: A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poirr ((𝑅 Po A B A) → ¬ B𝑅B)

Proof of Theorem poirr
StepHypRef Expression
1 df-3an 886 . . 3 ((B A B A B A) ↔ ((B A B A) B A))
2 anabs1 506 . . 3 (((B A B A) B A) ↔ (B A B A))
3 anidm 376 . . 3 ((B A B A) ↔ B A)
41, 2, 33bitrri 196 . 2 (B A ↔ (B A B A B A))
5 pocl 4031 . . . 4 (𝑅 Po A → ((B A B A B A) → (¬ B𝑅B ((B𝑅B B𝑅B) → B𝑅B))))
65imp 115 . . 3 ((𝑅 Po A (B A B A B A)) → (¬ B𝑅B ((B𝑅B B𝑅B) → B𝑅B)))
76simpld 105 . 2 ((𝑅 Po A (B A B A B A)) → ¬ B𝑅B)
84, 7sylan2b 271 1 ((𝑅 Po A B A) → ¬ B𝑅B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 884   wcel 1390   class class class wbr 3755   Po wpo 4022
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:  po2nr  4037  pofun  4040  sonr  4045  poirr2  4660  poxp  5794  swoer  6070
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