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Theorem elALT 4234
Description: Every set is an element of some other set. This has a shorter proof than el 4208 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 2804 . . 3  |-  x  e. 
_V
21snid 3680 . 2  |-  x  e. 
{ x }
3 snex 4232 . . 3  |-  { x }  e.  _V
4 eleq2 2357 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 2888 . 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 1531    e. wcel 1696   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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