Theorem pw0 3779

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 3499	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2320	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3617	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3638	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2235	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1372   {cab 2190   C_ wss 3165   (/)c0 3459   ~Pcpw 3615   {csn 3632

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638

This theorem is referenced by:  p0ex  4231  sn0topon  14478  sn0cld  14527