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Metamath Proof Explorer

Theorem dmv

Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003)

		Ref	Expression
	Assertion	dmv	$⊢ dom V = V$

Step	Hyp	Ref	Expression
1		ssv	$⊢ dom V \subseteq V$
2		dmi	$⊢ dom I = V$
3		ssv	$⊢ I \subseteq V$
4		dmss	$⊢ I \subseteq V \to dom I \subseteq dom V$
5	3 4	ax-mp	$⊢ dom I \subseteq dom V$
6	2 5	eqsstrri	$⊢ V \subseteq dom V$
7	1 6	eqssi	$⊢ dom V = V$