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Metamath Proof Explorer

Theorem pssv

Description: Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998)

		Ref	Expression
	Assertion	pssv	$⊢ A \subset V \leftrightarrow \neg A = V$

Step	Hyp	Ref	Expression
1		ssv	$⊢ A \subseteq V$
2		dfpss2	$⊢ A \subset V \leftrightarrow (A \subseteq V \land \neg A = V)$
3	1 2	mpbiran	$⊢ A \subset V \leftrightarrow \neg A = V$