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

Theorem elALT 4646
Description: Alternate proof of el 4585, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elALT  |-  E. y  x  e.  y
Distinct variable group:    x, y

Proof of Theorem elALT
StepHypRef Expression
1 vex 3081 . . 3  |-  x  e. 
_V
21snid 4016 . 2  |-  x  e. 
{ x }
3 snex 4644 . . 3  |-  { x }  e.  _V
4 eleq2 2527 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 3170 . 2  |-  ( x  e.  { x }  ->  E. y  x  e.  y )
62, 5ax-mp 5 1  |-  E. y  x  e.  y
Colors of variables: wff setvar class
Syntax hints:   E.wex 1587    e. wcel 1758   {csn 3988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3442  df-un 3444  df-nul 3749  df-sn 3989  df-pr 3991
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator