Theorem symrelcoss3

Description: The class of cosets by R is symmetric, see dfsymrel3 . (Contributed by Peter Mazsa, 28-Mar-2019) (Revised by Peter Mazsa, 17-Sep-2021)

	Ref	Expression
Assertion	symrelcoss3	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ≀ 𝑅 𝑦 → 𝑦 ≀ 𝑅 𝑥 ) ∧ Rel ≀ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		brcosscnvcoss	⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑥 ≀ 𝑅 𝑦 ↔ 𝑦 ≀ 𝑅 𝑥 ) )
2	1	el2v	⊢ ( 𝑥 ≀ 𝑅 𝑦 ↔ 𝑦 ≀ 𝑅 𝑥 )
3	2	biimpi	⊢ ( 𝑥 ≀ 𝑅 𝑦 → 𝑦 ≀ 𝑅 𝑥 )
4	3	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( 𝑥 ≀ 𝑅 𝑦 → 𝑦 ≀ 𝑅 𝑥 )
5		relcoss	⊢ Rel ≀ 𝑅
6	4 5	pm3.2i	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ≀ 𝑅 𝑦 → 𝑦 ≀ 𝑅 𝑥 ) ∧ Rel ≀ 𝑅 )