ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pw0 Unicode version

Theorem pw0 3505
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 3253 . . 3 
C_  (/)  (/)
21abbii 2153 . 2  {  |  C_  (/) }  {  |  (/) }
3 df-pw 3356 . 2  ~P (/)  {  |  C_  (/) }
4 df-sn 3376 . 2  { (/) }  {  |  (/) }
52, 3, 43eqtr4i 2070 1  ~P (/)  { (/)
}
Colors of variables: wff set class
Syntax hints:   wceq 1243   {cab 2026    C_ wss 2914   (/)c0 3221   ~Pcpw 3354   {csn 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3356  df-sn 3376
This theorem is referenced by:  p0ex  3933
  Copyright terms: Public domain W3C validator