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Theorem el 4583
 Description: Every set is an element of some other set. See elALT 4643 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 4580 . 2
2 ax9 1917 . . . . 5
32alrimiv 1781 . . . 4
4 ax8 1910 . . . 4
53, 4embantd 55 . . 3
65spimv 2114 . 2
71, 6eximii 1717 1
 Colors of variables: wff setvar class Syntax hints:   wi 4  wal 1450  wex 1671 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-pow 4579 This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676 This theorem is referenced by:  dtru  4592  dvdemo2  4636  axpownd  9044  zfcndinf  9061  domep  30510  distel  30521
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