Theorem pw0 4119

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 3764	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2567	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3953	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3969	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2483	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1444   {cab 2437   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-pw 3953  df-sn 3969

This theorem is referenced by:  p0ex  4590  pwfi  7869  ackbij1lem14  8663  fin1a2lem12  8841  0tsk  9180  hashbc  12616  incexclem  13894  sn0topon  20013  sn0cld  20106  ust0  21234  uhgra0v  25037  usgra0v  25098  esumnul  28869  rankeq1o  30938  ssoninhaus  31108  sge00  38218  uhgr0vb  39165  uhgr0  39166  uhg0v  39742