This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem snexg

Description: A singleton built on a set is a set. Special case of snex which does not require ax-nul and is intuitionistically valid. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 19-May-2013) Extract from snex and shorten proof. (Revised by BJ, 15-Jan-2025)

		Ref	Expression
	Assertion	snexg	$⊢ A \in V \to \{A\} \in V$

Proof

Step	Hyp	Ref	Expression
1		sneq	$⊢ x = A \to \{x\} = \{A\}$
2		vsnex	$⊢ \{x\} \in V$
3	1 2	eqeltrrdi	$⊢ x = A \to \{A\} \in V$
4	3	vtocleg	$⊢ A \in V \to \{A\} \in V$