Theorem pwnss 5293

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof shortened by BJ, 24-Jul-2025.)

Assertion

Ref	Expression
pwnss	⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Proof of Theorem pwnss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rru 3725	. . 3 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
2		ssel 3915	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴))
3	1, 2	mtoi 199	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴)
4		rabelpw 5277	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴)
5	3, 4	nsyl3 138	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114  {crab 3389   ⊆ wss 3889  𝒫 cpw 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543

This theorem is referenced by:  pwne  5294  pwuninel2  8224