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

Metamath Proof Explorer

Theorem 0elpw

Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007)

		Ref	Expression
	Assertion	0elpw	⊢ ∅ ∈ 𝒫 𝐴

Step	Hyp	Ref	Expression
1		0ss	⊢ ∅ ⊆ 𝐴
2		0ex	⊢ ∅ ∈ V
3	2	elpw	⊢ ( ∅ ∈ 𝒫 𝐴 ↔ ∅ ⊆ 𝐴 )
4	1 3	mpbir	⊢ ∅ ∈ 𝒫 𝐴