Theorem bj-snfromadj

Description: Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-snfromadj	⊢ { 𝑥 } ∈ V

Proof

Step	Hyp	Ref	Expression
1		0un	⊢ ( ∅ ∪ { 𝑥 } ) = { 𝑥 }
2		0ex	⊢ ∅ ∈ V
3		bj-adjg1	⊢ ( ∅ ∈ V → ( ∅ ∪ { 𝑥 } ) ∈ V )
4	2 3	ax-mp	⊢ ( ∅ ∪ { 𝑥 } ) ∈ V
5	1 4	eqeltrri	⊢ { 𝑥 } ∈ V