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Metamath Proof Explorer

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	$⊢ (\forall x \forall y (x ≀ R y \to y ≀ R x) \land Rel ≀ R)$

Proof

Step	Hyp	Ref	Expression
1		brcosscnvcoss	$⊢ (x \in V \land y \in V) \to (x ≀ R y \leftrightarrow y ≀ R x)$
2	1	el2v	$⊢ x ≀ R y \leftrightarrow y ≀ R x$
3	2	biimpi	$⊢ x ≀ R y \to y ≀ R x$
4	3	gen2	$⊢ \forall x \forall y (x ≀ R y \to y ≀ R x)$
5		relcoss	$⊢ Rel ≀ R$
6	4 5	pm3.2i	$⊢ (\forall x \forall y (x ≀ R y \to y ≀ R x) \land Rel ≀ R)$