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Theorem pwuninel2 5838
 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 ( A 𝑉 → ¬ 𝒫 A A)

Proof of Theorem pwuninel2
StepHypRef Expression
1 pwnss 3903 . 2 ( A 𝑉 → ¬ 𝒫 A A)
2 elssuni 3599 . 2 (𝒫 A A → 𝒫 A A)
31, 2nsyl 558 1 ( A 𝑉 → ¬ 𝒫 A A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351  ∪ cuni 3571 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-nel 2204  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572 This theorem is referenced by:  pnfnre  6864
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