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

Theorem 0el 3765
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 2885 . 2  |-  ( (/)  e.  A  <->  E. x  e.  A  x  =  (/) )
2 eq0 3763 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32rexbii 2862 . 2  |-  ( E. x  e.  A  x  =  (/)  <->  E. x  e.  A  A. y  -.  y  e.  x )
41, 3bitri 249 1  |-  ( (/)  e.  A  <->  E. x  e.  A  A. y  -.  y  e.  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   E.wrex 2800   (/)c0 3748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-v 3080  df-dif 3442  df-nul 3749
This theorem is referenced by:  axinf2  7961  zfinf2  7963  n0el  29175
  Copyright terms: Public domain W3C validator