Theorem pw0 3720

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 3448	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2282	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3561	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3582	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2196	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1343   {cab 2151   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   {csn 3576

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582

This theorem is referenced by:  p0ex  4167  sn0topon  12728  sn0cld  12777