Theorem pwne

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv . (Contributed by NM, 17-Nov-2008) (Proof shortened by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Assertion	pwne	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pwnss	⊢ ( 𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴 )
2		eqimss	⊢ ( 𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴 )
3	2	necon3bi	⊢ ( ¬ 𝒫 𝐴 ⊆ 𝐴 → 𝒫 𝐴 ≠ 𝐴 )
4	1 3	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴 )