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Metamath Proof Explorer

Theorem elrelscnveq4

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021)

		Ref	Expression
	Assertion	elrelscnveq4	$⊢ R \in Rels \to (R^{-1} \subseteq R \leftrightarrow \forall x \forall y (x R y \leftrightarrow y R x))$

Step	Hyp	Ref	Expression
1		elrelscnveq	$⊢ R \in Rels \to (R^{-1} \subseteq R \leftrightarrow R^{-1} = R)$
2		elrelscnveq2	$⊢ R \in Rels \to (R^{-1} = R \leftrightarrow \forall x \forall y (x R y \leftrightarrow y R x))$
3	1 2	bitrd	$⊢ R \in Rels \to (R^{-1} \subseteq R \leftrightarrow \forall x \forall y (x R y \leftrightarrow y R x))$