Theorem pw0 3846

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 3552	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2350	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3676	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3700	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2265	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1398   {cab 2220   C_ wss 3214   (/)c0 3512   ~Pcpw 3674   {csn 3694

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700

This theorem is referenced by:  p0ex  4306  hashfibc  11232  sn0topon  15079  sn0cld  15128  pw0ss  16204