Theorem pw0 3532

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 3283	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2194	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3384	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3404	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2111	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1284   {cab 2067   C_ wss 2973   (/)c0 3251   ~Pcpw 3382   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404

This theorem is referenced by:  p0ex  3959