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Theorem 0el 3484
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 2603 . 2  |-  ( (/)  e.  A  <->  E. x  e.  A  x  =  (/) )
2 eq0 3482 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32rexbii 2581 . 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 1530    = wceq 1632    e. wcel 1696   E.wrex 2557   (/)c0 3468
This theorem is referenced by:  axinf2  7357  zfinf2  7359  n0el  26828
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469
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