Theorem ersymtr 5932

Description: Equivalence relationship as symmetric, transitive relationship. (Contributed by SF, 22-Feb-2015.)

Assertion

Ref	Expression
ersymtr	⊢ (R Er A ↔ (R Sym A ∧ R Trans A))

Proof of Theorem ersymtr

Step	Hyp	Ref	Expression
1		df-er 5909	. . 3 ⊢ Er = ( Sym ∩ Trans )
2	1	breqi 4645	. 2 ⊢ (R Er A ↔ R( Sym ∩ Trans )A)
3		brin 4693	. 2 ⊢ (R( Sym ∩ Trans )A ↔ (R Sym A ∧ R Trans A))
4	2, 3	bitri 240	1 ⊢ (R Er A ↔ (R Sym A ∧ R Trans A))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∩ cin 3208   class class class wbr 4639   Trans ctrans 5888   Sym csym 5897   Er cer 5898

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-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-er 5909

This theorem is referenced by:  iserd  5942  ersym  5952  ertr  5954  ertrd  5955