Theorem pw0 3769

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 3490	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2312	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3607	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3628	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2227	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1364   {cab 2182   C_ wss 3157   (/)c0 3450   ~Pcpw 3605   {csn 3622

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628

This theorem is referenced by:  p0ex  4221  sn0topon  14324  sn0cld  14373