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Theorem elALT 4633
Description: Alternate proof of el 4575, 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 3061 . . 3  |-  x  e. 
_V
21snid 3999 . 2  |-  x  e. 
{ x }
3 snex 4631 . . 3  |-  { x }  e.  _V
4 eleq2 2475 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 3150 . 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 1633    e. wcel 1842   {csn 3971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3060  df-dif 3416  df-un 3418  df-nul 3738  df-sn 3972  df-pr 3974
This theorem is referenced by: (None)
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