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Theorem pw0 3763
Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
pw0  |-  ~P (/)  =  { (/)
}

Proof of Theorem pw0
StepHypRef Expression
1 ss0b 3485 . . 3  |-  ( x 
C_  (/)  <->  x  =  (/) )
21abbii 2396 . 2  |-  { x  |  x  C_  (/) }  =  { x  |  x  =  (/) }
3 df-pw 3628 . 2  |-  ~P (/)  =  {
x  |  x  C_  (/)
}
4 df-sn 3647 . 2  |-  { (/) }  =  { x  |  x  =  (/) }
52, 3, 43eqtr4i 2314 1  |-  ~P (/)  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    = wceq 1624   {cab 2270    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   {csn 3641
This theorem is referenced by:  p0ex  4196  pwfi  7146  ackbij1lem14  7854  fin1a2lem12  8032  0tsk  8372  hashbc  11385  incexclem  12289  sn0topon  16729  sn0cld  16821  rankeq1o  24208  ssoninhaus  24294
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-dif 3156  df-in 3160  df-ss 3167  df-nul 3457  df-pw 3628  df-sn 3647
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