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Theorem el 4164
Description: Every set is an element of some other set. (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 4161 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax-14 2144 . . . . 5  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1867 . . . 4  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax-13 2143 . . . 4  |-  ( z  =  x  ->  (
z  e.  y  ->  x  e.  y )
)
53, 4embantd 56 . . 3  |-  ( z  =  x  ->  (
( A. y ( y  e.  z  -> 
y  e.  x )  ->  z  e.  y )  ->  x  e.  y ) )
65spimv 1804 . 2  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
71, 6eximii 1595 1  |-  E. y  x  e.  y
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346   E.wex 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-13 2143  ax-14 2144  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  dtruarb  4177
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