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Theorem pwnss 5352
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 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Proof of Theorem pwnss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rru 3785 . . 3 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
2 ssel 3977 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
31, 2mtoi 199 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
4 rabelpw 5336 . 2 (𝐴𝑉 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
53, 4nsyl3 138 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602
This theorem is referenced by:  pwne  5353  pwuninel2  8299
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