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

Theorem pw0 3502
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 3250 . . 3 
C_  (/)  (/)
21abbii 2150 . 2  {  |  C_  (/) }  {  |  (/) }
3 df-pw 3353 . 2  ~P (/)  {  |  C_  (/) }
4 df-sn 3373 . 2  { (/) }  {  |  (/) }
52, 3, 43eqtr4i 2067 1  ~P (/)  { (/)
}
Colors of variables: wff set class
Syntax hints:   wceq 1242   {cab 2023    C_ wss 2911   (/)c0 3218   ~Pcpw 3351   {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373
This theorem is referenced by:  p0ex  3930
  Copyright terms: Public domain W3C validator