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Theorem pwne0 4459
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 3638 . 2  |-  -.  (/)  e.  (/)
2 0elpw 4458 . . . 4  |-  (/)  e.  ~P A
3 eleq2 2502 . . . 4  |-  ( ~P A  =  (/)  ->  ( (/) 
e.  ~P A  <->  (/)  e.  (/) ) )
42, 3mpbii 211 . . 3  |-  ( ~P A  =  (/)  ->  (/)  e.  (/) )
54necon3bi 2650 . 2  |-  ( -.  (/)  e.  (/)  ->  ~P A  =/=  (/) )
61, 5ax-mp 5 1  |-  ~P A  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1364    e. wcel 1761    =/= wne 2604   (/)c0 3634   ~Pcpw 3857
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 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-nul 4418
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-v 2972  df-dif 3328  df-in 3332  df-ss 3339  df-nul 3635  df-pw 3859
This theorem is referenced by:  undefne0  6794
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