Theorem pw0 4131

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 3778	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2588	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3973	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3989	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2493	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1370   {cab 2439   C_ wss 3439   (/)c0 3748   ~Pcpw 3971   {csn 3988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749  df-pw 3973  df-sn 3989

This theorem is referenced by:  p0ex  4590  pwfi  7720  ackbij1lem14  8516  fin1a2lem12  8694  0tsk  9036  hashbc  12327  incexclem  13420  sn0topon  18737  sn0cld  18829  ust0  19929  uhgra0v  23416  usgra0v  23462  esumnul  26667  rankeq1o  28373  ssoninhaus  28458