Theorem eqvreltrrel 36713

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 36698	. 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
2	1	simp3bi 1146	1 ⊢ ( EqvRel 𝑅 → TrRel 𝑅)

Syntax hints:   → wi 4   RefRel wrefrel 36339   SymRel wsymrel 36345   TrRel wtrrel 36348   EqvRel weqvrel 36350

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 206  df-an 397  df-3an 1088  df-eqvrel 36698

This theorem is referenced by: (None)