Theorem el 3922

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 3919	. 2 |- E.yA.z(A.y(y e. z -> y e. x) -> z e. y)
2		ax-14 1402	. . . . 5 |- (z = x -> (y e. z -> y e. x))
3	2	alrimiv 1751	. . . 4 |- (z = x -> A.y(y e. z -> y e. x))
4		ax-13 1401	. . . 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 1689	. 2 |- (A.z(A.y(y e. z -> y e. x) -> z e. y) -> x e. y)
7	1, 6	eximii 1490	1 |- E.y x e. y

Syntax hints:   -> wi 4  A.wal 1240  E.wex 1378

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  dtruarb  3933