Theorem pw0 3765

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 3486	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2309	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3603	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3624	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2224	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1364   {cab 2179   C_ wss 3153   (/)c0 3446   ~Pcpw 3601   {csn 3618

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624

This theorem is referenced by:  p0ex  4217  sn0topon  14256  sn0cld  14305