Theorem pwne 5371

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4928. (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 5370	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
2		eqimss 4067	. . 3 ⊢ (𝒫 𝐴 = 𝐴 → 𝒫 𝐴 ⊆ 𝐴)
3	2	necon3bi 2973	. 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 2108   ≠ wne 2946   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624

This theorem is referenced by:  pnfnemnf  11345