ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwuninel2 GIF version

Theorem pwuninel2 5858
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 3908 . 2 ( 𝐴𝑉 → ¬ 𝒫 𝐴 𝐴)
2 elssuni 3604 . 2 (𝒫 𝐴𝐴 → 𝒫 𝐴 𝐴)
31, 2nsyl 558 1 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1393  wss 2914  𝒫 cpw 3356   cuni 3576
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3871
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-rab 2312  df-v 2556  df-in 2921  df-ss 2928  df-pw 3358  df-uni 3577
This theorem is referenced by:  pnfnre  7009
  Copyright terms: Public domain W3C validator