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

Step	Hyp	Ref	Expression
1		brin 4693	. . 3 ⊢ (R(( Ref ∩ Trans ) ∩ Antisym )A ↔ (R( Ref ∩ Trans )A ∧ R Antisym A))
2		brin 4693	. . . 4 ⊢ (R( Ref ∩ Trans )A ↔ (R Ref A ∧ R Trans A))
3	2	anbi1i 676	. . 3 ⊢ ((R( Ref ∩ Trans )A ∧ R Antisym A) ↔ ((R Ref A ∧ R Trans A) ∧ R Antisym A))
4	1, 3	bitri 240	. 2 ⊢ (R(( Ref ∩ Trans ) ∩ Antisym )A ↔ ((R Ref A ∧ R Trans A) ∧ R Antisym A))
5		df-partial 5902	. . 3 ⊢ Po = (( Ref ∩ Trans ) ∩ Antisym )
6	5	breqi 4645	. 2 ⊢ (R Po A ↔ R(( Ref ∩ Trans ) ∩ 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))
8	4, 6, 7	3bitr4i 268	1 ⊢ (R Po A ↔ (R Ref A ∧ R Trans A ∧ R Antisym A))

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