Theorem pw0 3820

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 3534	. . 3 |- ( x C_ (/) <-> x = (/) )
2	1	abbii 2347	. 2 |- { x | x C_ (/) } = { x | x = (/) }
3		df-pw 3654	. 2 |- ~P (/) = { x | x C_ (/) }
4		df-sn 3675	. 2 |- { (/) } = { x | x = (/) }
5	2, 3, 4	3eqtr4i 2262	1 |- ~P (/) = { (/) }

Syntax hints:   = wceq 1397   {cab 2217   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675

This theorem is referenced by:  p0ex  4278  sn0topon  14811  sn0cld  14860  pw0ss  15933