Theorem pwne 5242

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4816. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Assertion

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

Proof of Theorem pwne

Step	Hyp	Ref	Expression
1		pwnss 5241	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
2		eqimss 3957	. . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴)
3	2	necon3bi 2967	. 2 ⊢ (¬ 𝒫 𝐴 ⊆ 𝐴 → 𝒫 𝐴 ≠ 𝐴)
4	1, 3	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110   ≠ wne 2940   ⊆ wss 3866  𝒫 cpw 4513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-pw 4515

This theorem is referenced by:  pnfnemnf  10888