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Theorem 0elpw 4088
Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
Assertion
Ref Expression
0elpw  |-  (/)  e.  ~P A

Proof of Theorem 0elpw
StepHypRef Expression
1 0ss 3401 . 2  |-  (/)  C_  A
2 0ex 4055 . . 3  |-  (/)  e.  _V
32elpw 3516 . 2  |-  ( (/)  e.  ~P A  <->  (/)  C_  A
)
41, 3mpbir 145 1  |-  (/)  e.  ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 1480    C_ wss 3071   (/)c0 3363   ~Pcpw 3510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512
This theorem is referenced by:  ordpwsucexmid  4485  pw1nct  13282
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