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Theorem porta 5933
 Description: Partial ordering as reflexive, transitive, antisymmetric relationship. (Contributed by SF, 12-Mar-2015.)
Assertion
Ref Expression
porta (R Po A ↔ (R Ref A R Trans A R Antisym A))

Proof of Theorem porta
StepHypRef Expression
1 brin 4693 . . 3 (R(( RefTrans ) ∩ Antisym )A ↔ (R( RefTrans )A R Antisym A))
2 brin 4693 . . . 4 (R( RefTrans )A ↔ (R Ref A R Trans A))
32anbi1i 676 . . 3 ((R( RefTrans )A R Antisym A) ↔ ((R Ref A R Trans A) R Antisym A))
41, 3bitri 240 . 2 (R(( RefTrans ) ∩ Antisym )A ↔ ((R Ref A R Trans A) R Antisym A))
5 df-partial 5902 . . 3 Po = (( RefTrans ) ∩ Antisym )
65breqi 4645 . 2 (R Po AR(( RefTrans ) ∩ Antisym )A)
7 df-3an 936 . 2 ((R Ref A R Trans A R Antisym A) ↔ ((R Ref A R Trans A) R Antisym A))
84, 6, 73bitr4i 268 1 (R Po A ↔ (R Ref A R Trans A R Antisym A))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934   ∩ cin 3208   class class class wbr 4639   Trans ctrans 5888   Ref cref 5889   Antisym cantisym 5890   Po cpartial 5891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-br 4640  df-partial 5902 This theorem is referenced by:  pod  5936  weds  5938  nchoicelem19  6307
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