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Theorem pwnss 5141
 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 3704 . . 3 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
2 ssel 3883 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
31, 2mtoi 200 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
4 ssrab2 3977 . . 3 {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴
5 elpw2g 5138 . . 3 (𝐴𝑉 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴))
64, 5mpbiri 259 . 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 2081  {crab 3109   ⊆ wss 3859  𝒫 cpw 4453 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-in 3866  df-ss 3874  df-pw 4455 This theorem is referenced by:  pwne  5142  pwuninel2  7791
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