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Theorem 0elpw 4148
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 3452 . 2  |-  (/)  C_  A
2 0ex 4114 . . 3  |-  (/)  e.  _V
32elpw 3570 . 2  |-  ( (/)  e.  ~P A  <->  (/)  C_  A
)
41, 3mpbir 145 1  |-  (/)  e.  ~P A
Colors of variables: wff set class
Syntax hints:    e. wcel 2141    C_ wss 3121   (/)c0 3414   ~Pcpw 3564
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4113
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566
This theorem is referenced by:  ordpwsucexmid  4552  pw1on  7192  pw1ne0  7194  pw1nct  13998
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