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Metamath Proof Explorer

Theorem vss

Description: Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011)

		Ref	Expression
	Assertion	vss	$⊢ V \subseteq A \leftrightarrow A = V$

Proof

Step	Hyp	Ref	Expression
1		ssv	$⊢ A \subseteq V$
2	1	biantrur	$⊢ V \subseteq A \leftrightarrow (A \subseteq V \land V \subseteq A)$
3		eqss	$⊢ A = V \leftrightarrow (A \subseteq V \land V \subseteq A)$
4	2 3	bitr4i	$⊢ V \subseteq A \leftrightarrow A = V$