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.

Step	Hyp	Ref	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 ) )
3	2	alrimiv 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 ) )
5	3, 4	embantd 56	. . 3 |- ( z = x -> ( ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y ) )
6	5	spimv 1804	. 2 |- ( A. z ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y )
7	1, 6	eximii 1595	1 |- E. y x e. y

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