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Theorem elALT 4399
Description: Every set is an element of some other set. This has a shorter proof than el 4373 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 2951 . . 3  |-  x  e. 
_V
21snid 3833 . 2  |-  x  e. 
{ x }
3 snex 4397 . . 3  |-  { x }  e.  _V
4 eleq2 2496 . . 3  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
53, 4spcev 3035 . 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 1550    e. wcel 1725   {csn 3806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-pr 3813
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