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Metamath Proof Explorer

Theorem dfpss2

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996)

		Ref	Expression
	Assertion	dfpss2	$⊢ A \subset B \leftrightarrow (A \subseteq B \land \neg A = B)$

Step	Hyp	Ref	Expression
1		df-pss	$⊢ A \subset B \leftrightarrow (A \subseteq B \land A \neq B)$
2		df-ne	$⊢ A \neq B \leftrightarrow \neg A = B$
3	2	anbi2i	$⊢ (A \subseteq B \land A \neq B) \leftrightarrow (A \subseteq B \land \neg A = B)$
4	1 3	bitri	$⊢ A \subset B \leftrightarrow (A \subseteq B \land \neg A = B)$