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Theorem elALT 4348
Description: Every set is an element of some other set. This has a shorter proof than el 4322 but uses 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 2902 . . 3  |-  x  e. 
_V
21snid 3784 . 2  |-  x  e. 
{ x }
3 snex 4346 . . 3  |-  { x }  e.  _V
4 eleq2 2448 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 2986 . 2  |-  ( x  e.  { x }  ->  E. y  x  e.  y )
62, 5ax-mp 8 1  |-  E. y  x  e.  y
Colors of variables: wff set class
Syntax hints:   E.wex 1547    e. wcel 1717   {csn 3757
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-nul 3572  df-sn 3763  df-pr 3764
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