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Theorem pwne0 4607
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 noel 3774 . 2  |-  -.  (/)  e.  (/)
2 0elpw 4606 . . . 4  |-  (/)  e.  ~P A
3 eleq2 2516 . . . 4  |-  ( ~P A  =  (/)  ->  ( (/) 
e.  ~P A  <->  (/)  e.  (/) ) )
42, 3mpbii 211 . . 3  |-  ( ~P A  =  (/)  ->  (/)  e.  (/) )
54necon3bi 2672 . 2  |-  ( -.  (/)  e.  (/)  ->  ~P A  =/=  (/) )
61, 5ax-mp 5 1  |-  ~P A  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804    =/= wne 2638   (/)c0 3770   ~Pcpw 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3771  df-pw 3999
This theorem is referenced by:  undefne0  7010
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