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Theorem el 4583
Description: Every set is an element of some other set. See elALT 4643 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
el  |-  E. y  x  e.  y
Distinct variable group:    x, y

Proof of Theorem el
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 zfpow 4580 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax9 1917 . . . . 5  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1781 . . . 4  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax8 1910 . . . 4  |-  ( z  =  x  ->  (
z  e.  y  ->  x  e.  y )
)
53, 4embantd 55 . . 3  |-  ( z  =  x  ->  (
( A. y ( y  e.  z  -> 
y  e.  x )  ->  z  e.  y )  ->  x  e.  y ) )
65spimv 2114 . 2  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
71, 6eximii 1717 1  |-  E. y  x  e.  y
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-pow 4579
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  dtru  4592  dvdemo2  4636  axpownd  9044  zfcndinf  9061  domep  30510  distel  30521
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