MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwne0 Structured version   Visualization version   GIF version

Theorem pwne0 5253
Description: A power class is never empty. (Contributed by NM, 3-Sep-2018.)
Assertion
Ref Expression
pwne0 𝒫 𝐴 ≠ ∅

Proof of Theorem pwne0
StepHypRef Expression
1 0elpw 5252 . 2 ∅ ∈ 𝒫 𝐴
21ne0ii 4306 1 𝒫 𝐴 ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 3020  c0 4294  𝒫 cpw 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543
This theorem is referenced by:  undefne0  7939  afv20defat  43294
  Copyright terms: Public domain W3C validator