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Theorem pwnss 5223
 Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwnss (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Proof of Theorem pwnss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rru 3696 . . 3 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
2 ssel 3888 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
31, 2mtoi 202 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
4 ssrab2 3987 . . 3 {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴
5 elpw2g 5219 . . 3 (𝐴𝑉 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴))
64, 5mpbiri 261 . 2 (𝐴𝑉 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
73, 6nsyl3 140 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2112  {crab 3075   ⊆ wss 3861  𝒫 cpw 4498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3080  df-v 3412  df-in 3868  df-ss 3878  df-pw 4500 This theorem is referenced by:  pwne  5224  pwuninel2  7957
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