Theorem eqvreltrrel 38542

Description: An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)

Assertion

Ref	Expression
eqvreltrrel	⊢ ( EqvRel 𝑅 → TrRel 𝑅)

Proof of Theorem eqvreltrrel

Step	Hyp	Ref	Expression
1		df-eqvrel 38527	. 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
2	1	simp3bi 1147	1 ⊢ ( EqvRel 𝑅 → TrRel 𝑅)

Syntax hints:   → wi 4   RefRel wrefrel 38129   SymRel wsymrel 38135   TrRel wtrrel 38138   EqvRel weqvrel 38140

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 207  df-an 396  df-3an 1088  df-eqvrel 38527

This theorem is referenced by: (None)