Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvrelsymrel Structured version   Visualization version   GIF version

Theorem eqvrelsymrel 36335
Description: An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
Assertion
Ref Expression
eqvrelsymrel ( EqvRel 𝑅 → SymRel 𝑅)

Proof of Theorem eqvrelsymrel
StepHypRef Expression
1 df-eqvrel 36321 . 2 ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
21simp2bi 1147 1 ( EqvRel 𝑅 → SymRel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   RefRel wrefrel 35962   SymRel wsymrel 35968   TrRel wtrrel 35971   EqvRel weqvrel 35973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-eqvrel 36321
This theorem is referenced by:  eqvrelim  36337  eqvrelsym  36341
  Copyright terms: Public domain W3C validator