Theorem pwsn 4428

Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.)

Assertion

Ref	Expression
pwsn	|- ~P { A } = { (/) , { A } }

Proof of Theorem pwsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssn 4358	. . 3 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
2	1	abbii 2739	. 2 |- { x | x C_ { A } } = { x | ( x = (/) \/ x = { A } ) }
3		df-pw 4160	. 2 |- ~P { A } = { x | x C_ { A } }
4		dfpr2 4195	. 2 |- { (/) , { A } } = { x | ( x = (/) \/ x = { A } ) }
5	2, 3, 4	3eqtr4i 2654	1 |- ~P { A } = { (/) , { A } }

Syntax hints:   \/ wo 383   = wceq 1483   {cab 2608   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by:  pmtrsn  17939  topsn  20735  conncompid  21234  lfuhgr1v0e  26146  esumsnf  30126  cvmlift2lem9  31293  rrxtopn0b  40516  sge0sn  40596