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Theorem pwuninel2 6179
 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 4083 . 2 ( 𝐴𝑉 → ¬ 𝒫 𝐴 𝐴)
2 elssuni 3764 . 2 (𝒫 𝐴𝐴 → 𝒫 𝐴 𝐴)
31, 2nsyl 617 1 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1480   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737 This theorem is referenced by:  pnfnre  7819
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