Theorem el 4634

Description: Every set is an element of some other set. See elALT 4695 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

Distinct variable group:   ,

Proof of Theorem el

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpow 4631	. 2
2		ax-9 1822	. . . . 5
3	2	alrimiv 1719	. . . 4
4		ax-8 1820	. . . 4
5	3, 4	embantd 54	. . 3
6	5	spimv 2009	. 2
7	1, 6	eximii 1658	1

Syntax hints:  ->wi 4  A.wal 1393  E.wex 1612

This theorem is referenced by:  dtru  4643  dvdemo2  4688  axpownd  8999  zfcndinf  9017  domep  29225  distel  29236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-pow 4630

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617