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Theorem el 4208
Description: Every set is an element of some other set. See elALT 4234 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 4205 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax-14 1700 . . . . . 6  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1621 . . . . 5  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax-13 1698 . . . . 5  |-  ( z  =  x  ->  (
z  e.  y  ->  x  e.  y )
)
53, 4embantd 50 . . . 4  |-  ( z  =  x  ->  (
( A. y ( y  e.  z  -> 
y  e.  x )  ->  z  e.  y )  ->  x  e.  y ) )
65spimv 1943 . . 3  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
76eximi 1566 . 2  |-  ( E. y A. z ( A. y ( y  e.  z  ->  y  e.  x )  ->  z  e.  y )  ->  E. y  x  e.  y )
81, 7ax-mp 8 1  |-  E. y  x  e.  y
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  dtru  4217  dvdemo2  4227  axpownd  8239  zfcndinf  8256  domep  24220  distel  24231
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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