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Theorem 0el 3755
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 2931 . 2  |-  ( (/)  e.  A  <->  E. x  e.  A  x  =  (/) )
2 eq0 3753 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32rexbii 2905 . 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 1403    = wceq 1405    e. wcel 1842   E.wrex 2754   (/)c0 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rex 2759  df-v 3060  df-dif 3416  df-nul 3738
This theorem is referenced by:  axinf2  8010  zfinf2  8012  n0el  31863
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