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Metamath Proof Explorer

Theorem dmsnop

Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Revised by Mario Carneiro, 26-Apr-2015)

		Ref	Expression
	Hypothesis	dmsnop.1	$⊢ B \in V$
	Assertion	dmsnop	$⊢ dom \{⟨A, B⟩\} = \{A\}$

Proof

Step	Hyp	Ref	Expression
1		dmsnop.1	$⊢ B \in V$
2		dmsnopg	$⊢ B \in V \to dom \{⟨A, B⟩\} = \{A\}$
3	1 2	ax-mp	$⊢ dom \{⟨A, B⟩\} = \{A\}$