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Theorem 0el 3589
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 2698 . 2  |-  ( (/)  e.  A  <->  E. x  e.  A  x  =  (/) )
2 eq0 3587 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32rexbii 2676 . 2  |-  ( E. x  e.  A  x  =  (/)  <->  E. x  e.  A  A. y  -.  y  e.  x )
41, 3bitri 241 1  |-  ( (/)  e.  A  <->  E. x  e.  A  A. y  -.  y  e.  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   E.wrex 2652   (/)c0 3573
This theorem is referenced by:  axinf2  7530  zfinf2  7532  n0el  26401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rex 2657  df-v 2903  df-dif 3268  df-nul 3574
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