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Theorem el 4005
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 4002 . 2  |-  E. y A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )
2 ax-14 1450 . . . . 5  |-  ( z  =  x  ->  (
y  e.  z  -> 
y  e.  x ) )
32alrimiv 1802 . . . 4  |-  ( z  =  x  ->  A. y
( y  e.  z  ->  y  e.  x
) )
4 ax-13 1449 . . . 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 1739 . 2  |-  ( A. z ( A. y
( y  e.  z  ->  y  e.  x
)  ->  z  e.  y )  ->  x  e.  y )
71, 6eximii 1538 1  |-  E. y  x  e.  y
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  dtruarb  4017
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