Theorem pw0 3857

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 3562	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2354	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3687	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3711	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2269	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1402   {cab 2224   C_ wss 3220   (/)c0 3520   ~Pcpw 3685   {csn 3705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711

This theorem is referenced by:  p0ex  4320  hashfibc  11261  sn0topon  15112  sn0cld  15161  pw0ss  16238