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Theorem pwnss 5345
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 3773 . . 3 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
2 ssel 3971 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
31, 2mtoi 198 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
4 ssrab2 4073 . . 3 {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴
5 elpw2g 5340 . . 3 (𝐴𝑉 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴))
64, 5mpbiri 258 . 2 (𝐴𝑉 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
73, 6nsyl3 138 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2099  {crab 3428  wss 3945  𝒫 cpw 4598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-in 3952  df-ss 3962  df-pw 4600
This theorem is referenced by:  pwne  5346  pwuninel2  8273
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