MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0el Unicode version

Theorem 0el 3471
Description: Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
0el  |-  ( (/)  e.  A  <->  E. x  e.  A  A. y  -.  y  e.  x )
Distinct variable groups:    x, A    x, y
Allowed substitution hint:    A( y)

Proof of Theorem 0el
StepHypRef Expression
1 risset 2590 . 2  |-  ( (/)  e.  A  <->  E. x  e.  A  x  =  (/) )
2 eq0 3469 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32rexbii 2568 . 2  |-  ( E. x  e.  A  x  =  (/)  <->  E. x  e.  A  A. y  -.  y  e.  x )
41, 3bitri 240 1  |-  ( (/)  e.  A  <->  E. x  e.  A  A. y  -.  y  e.  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  axinf2  7341  zfinf2  7343  n0el  26725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
  Copyright terms: Public domain W3C validator