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Metamath Proof Explorer

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	$⊢ A \in V \to 𝒫 A \neq A$

Proof

Step	Hyp	Ref	Expression
1		pwnss	$⊢ A \in V \to \neg 𝒫 A \subseteq A$
2		eqimss	$⊢ 𝒫 A = A \to 𝒫 A \subseteq A$
3	2	necon3bi	$⊢ \neg 𝒫 A \subseteq A \to 𝒫 A \neq A$
4	1 3	syl	$⊢ A \in V \to 𝒫 A \neq A$