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Theorem elALT 4218
Description: Every set is an element of some other set. This has a shorter proof than el 4192 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 2791 . . 3  |-  x  e. 
_V
21snid 3667 . 2  |-  x  e. 
{ x }
3 snex 4216 . . 3  |-  { x }  e.  _V
4 eleq2 2344 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 2875 . 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 1528    e. wcel 1684   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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