Theorem pw0 4180

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 3820	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2601	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 4018	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 4034	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2506	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1379   {cab 2452   C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034

This theorem is referenced by:  p0ex  4640  pwfi  7827  ackbij1lem14  8625  fin1a2lem12  8803  0tsk  9145  hashbc  12483  incexclem  13628  sn0topon  19367  sn0cld  19459  ust0  20590  uhgra0v  24133  usgra0v  24194  esumnul  27884  rankeq1o  29755  ssoninhaus  29840  uhg0v  32167