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Metamath Proof Explorer

Theorem idinxpssinxp4

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 ). (Contributed by Peter Mazsa, 8-Mar-2019)

		Ref	Expression
	Assertion	idinxpssinxp4	$⊢ \forall x \in A \forall y \in A (x = y \to x R y) \leftrightarrow \forall x \in A x R x$

Proof

Step	Hyp	Ref	Expression
1		idinxpssinxp	$⊢ I \cap (A \times A) \subseteq R \cap (A \times A) \leftrightarrow \forall x \in A \forall y \in A (x = y \to x R y)$
2		idinxpssinxp2	$⊢ I \cap (A \times A) \subseteq R \cap (A \times A) \leftrightarrow \forall x \in A x R x$
3	1 2	bitr3i	$⊢ \forall x \in A \forall y \in A (x = y \to x R y) \leftrightarrow \forall x \in A x R x$