Theorem symrelcoss

Description: The class of cosets by R is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021)

	Ref	Expression
Assertion	symrelcoss	⊢ SymRel ≀ 𝑅

Proof

Step	Hyp	Ref	Expression
1		symrelcoss2	⊢ ( ◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅 )
2		dfsymrel2	⊢ ( SymRel ≀ 𝑅 ↔ ( ◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅 ) )
3	1 2	mpbir	⊢ SymRel ≀ 𝑅