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Theorem pwnss 5241
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 3767 . . 3 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
2 ssel 3958 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
31, 2mtoi 200 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝒫 𝐴)
4 ssrab2 4053 . . 3 {𝑥𝐴 ∣ ¬ 𝑥𝑥} ⊆ 𝐴
5 elpw2g 5238 . . 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 2105  {crab 3139  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  pwne  5242  pwuninel2  7929
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