Theorem pw0 3786

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 3504	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2322	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3623	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3644	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2237	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1373   {cab 2192   C_ wss 3170   (/)c0 3464   ~Pcpw 3621   {csn 3638

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644

This theorem is referenced by:  p0ex  4240  sn0topon  14635  sn0cld  14684  pw0ss  15754