Theorem el 3931

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 3928	. 2 |- E. y A. z ( A. y ( y e. z -> y e. x ) -> z e. y )
2		ax-14 1405	. . . . 5 |- ( z = x -> ( y e. z -> y e. x ) )
3	2	alrimiv 1754	. . . 4 |- ( z = x -> A. y ( y e. z -> y e. x ) )
4		ax-13 1404	. . . 4 |- ( z = x -> ( z e. y -> x e. y ) )
5	3, 4	embantd 50	. . 3 |- ( z = x -> ( ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y ) )
6	5	spimv 1692	. 2 |- ( A. z ( A. y ( y e. z -> y e. x ) -> z e. y ) -> x e. y )
7	1, 6	eximii 1493	1 |- E. y x e. y

Syntax hints:   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  dtruarb  3942