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Theorem el 4373
Description: Every set is an element of some other set. See elALT 4399 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 4370 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax-14 1729 . . . . 5  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1641 . . . 4  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax-13 1727 . . . 4  |-  ( z  =  x  ->  (
z  e.  y  ->  x  e.  y )
)
53, 4embantd 52 . . 3  |-  ( z  =  x  ->  (
( A. y ( y  e.  z  -> 
y  e.  x )  ->  z  e.  y )  ->  x  e.  y ) )
65spimv 1963 . 2  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
71, 6eximii 1587 1  |-  E. y  x  e.  y
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  dtru  4382  dvdemo2  4392  axpownd  8468  zfcndinf  8485  domep  25412  distel  25423
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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