Theorem el 4847

Description: Every set is an element of some other set. See elALT 4910 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.

Step	Hyp	Ref	Expression
1		zfpow 4844	. 2 |- E. y A. z ( A. y ( y e. z -> y e. x ) -> z e. y )
2		ax9 2003	. . . . 5 |- ( z = x -> ( y e. z -> y e. x ) )
3	2	alrimiv 1855	. . . 4 |- ( z = x -> A. y ( y e. z -> y e. x ) )
4		ax8 1996	. . . 4 |- ( z = x -> ( z e. y -> x e. y ) )
5	3, 4	embantd 59	. . 3 |- ( z = x -> ( ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y ) )
6	5	spimv 2257	. 2 |- ( A. z ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y )
7	1, 6	eximii 1764	1 |- E. y x e. y

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-11 2034  ax-12 2047  ax-13 2246  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  dtru  4857  dvdemo2  4903  axpownd  9423  zfcndinf  9440  domep  31698  distel  31709