Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  el Unicode version

Theorem el 4208
 Description: Every set is an element of some other set. See elALT 4234 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 4205 . 2
2 ax-14 1700 . . . . . 6
32alrimiv 1621 . . . . 5
4 ax-13 1698 . . . . 5
53, 4embantd 50 . . . 4
65spimv 1943 . . 3
76eximi 1566 . 2
81, 7ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1530  wex 1531   wceq 1632   wcel 1696 This theorem is referenced by:  dtru  4217  dvdemo2  4227  axpownd  8239  zfcndinf  8256  domep  24220  distel  24231 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-pow 4204 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
 Copyright terms: Public domain W3C validator