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Theorem pwne0 4610
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 3782 . 2  |-  -.  (/)  e.  (/)
2 0elpw 4609 . . . 4  |-  (/)  e.  ~P A
3 eleq2 2533 . . . 4  |-  ( ~P A  =  (/)  ->  ( (/) 
e.  ~P A  <->  (/)  e.  (/) ) )
42, 3mpbii 211 . . 3  |-  ( ~P A  =  (/)  ->  (/)  e.  (/) )
54necon3bi 2689 . 2  |-  ( -.  (/)  e.  (/)  ->  ~P A  =/=  (/) )
61, 5ax-mp 5 1  |-  ~P A  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762    =/= wne 2655   (/)c0 3778   ~Pcpw 4003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005
This theorem is referenced by:  undefne0  6998
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