Theorem pw0 3703

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 3433	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2273	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3545	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3566	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2188	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1335   {cab 2143   C_ wss 3102   (/)c0 3394   ~Pcpw 3543   {csn 3560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566

This theorem is referenced by:  p0ex  4149  sn0topon  12475  sn0cld  12524