Theorem el 4157

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 4154	. 2 |- E. y A. z ( A. y ( y e. z -> y e. x ) -> z e. y )
2		ax-14 2139	. . . . 5 |- ( z = x -> ( y e. z -> y e. x ) )
3	2	alrimiv 1862	. . . 4 |- ( z = x -> A. y ( y e. z -> y e. x ) )
4		ax-13 2138	. . . 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 1799	. 2 |- ( A. z ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y )
7	1, 6	eximii 1590	1 |- E. y x e. y

Syntax hints:   -> wi 4   A.wal 1341   E.wex 1480

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-13 2138  ax-14 2139  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  dtruarb  4170