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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.
StepHypRef Expression
1 zfpow 4631 . 2
2 ax-9 1822 . . . . 5
32alrimiv 1719 . . . 4
4 ax-8 1820 . . . 4
53, 4embantd 54 . . 3
65spimv 2009 . 2
71, 6eximii 1658 1
 Colors of variables: wff setvar class 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
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