Theorem pw0 3667

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

Step	Hyp	Ref	Expression
1		ss0b 3402	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2255	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3512	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3533	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2170	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1331   {cab 2125   C_ wss 3071   (/)c0 3363   ~Pcpw 3510   {csn 3527

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

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  df-sn 3533

This theorem is referenced by:  p0ex  4112  sn0topon  12257  sn0cld  12306