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Theorem pwuninel2 6061
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwuninel2 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Proof of Theorem pwuninel2
StepHypRef Expression
1 pwnss 4000 . 2 ( 𝐴𝑉 → ¬ 𝒫 𝐴 𝐴)
2 elssuni 3687 . 2 (𝒫 𝐴𝐴 → 𝒫 𝐴 𝐴)
31, 2nsyl 594 1 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1439  wss 3000  𝒫 cpw 3433   cuni 3659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-nel 2352  df-rab 2369  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-uni 3660
This theorem is referenced by:  pnfnre  7583
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