MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snexALT Unicode version

Theorem snexALT 4328
Description: A singleton is a set. Theorem 7.13 of [Quine] p. 51, but proved using only Extensionality, Power Set, and Separation. Unlike the proof of zfpair 4344, Replacement is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) See also snex 4348. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snexALT  |-  { A }  e.  _V

Proof of Theorem snexALT
StepHypRef Expression
1 snsspw 3914 . . 3  |-  { A }  C_  ~P A
2 ssexg 4292 . . 3  |-  ( ( { A }  C_  ~P A  /\  ~P A  e.  _V )  ->  { A }  e.  _V )
31, 2mpan 652 . 2  |-  ( ~P A  e.  _V  ->  { A }  e.  _V )
4 pwexg 4326 . . . 4  |-  ( A  e.  _V  ->  ~P A  e.  _V )
54con3i 129 . . 3  |-  ( -. 
~P A  e.  _V  ->  -.  A  e.  _V )
6 snprc 3816 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 187 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
8 0ex 4282 . . . 4  |-  (/)  e.  _V
97, 8syl6eqel 2477 . . 3  |-  ( -.  A  e.  _V  ->  { A }  e.  _V )
105, 9syl 16 . 2  |-  ( -. 
~P A  e.  _V  ->  { A }  e.  _V )
113, 10pm2.61i 158 1  |-  { A }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2901    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   {csn 3759
This theorem is referenced by:  p0exALT  4330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-nul 3574  df-pw 3746  df-sn 3765
  Copyright terms: Public domain W3C validator