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Theorem eq0 3557
Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
eq0  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem eq0
StepHypRef Expression
1 neq0 3553 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 df-ex 1547 . . 3  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
31, 2bitri 240 . 2  |-  ( -.  A  =  (/)  <->  -.  A. x  -.  x  e.  A
)
43con4bii 288 1  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1545   E.wex 1546    = wceq 1647    e. wcel 1715   (/)c0 3543
This theorem is referenced by:  0el  3559  ssdif0  3601  difin0ss  3609  inssdif0  3610  ralf0  3649  disjiun  4115  0ex  4252  dm0  4995  reldm0  4999  uzwo  10432  uzwoOLD  10433  fzouzdisj  11059  hashgt0elex  11557  hausdiag  17556  rnelfmlem  17860  nninfnub  26053  prtlem14  26333  stoweidlem34  27374  stoweidlem44  27384  bnj1476  28631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-v 2875  df-dif 3241  df-nul 3544
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