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Theorem eq0 3634
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 3630 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
2 df-ex 1551 . . 3  |-  ( E. x  x  e.  A  <->  -. 
A. x  -.  x  e.  A )
31, 2bitri 241 . 2  |-  ( -.  A  =  (/)  <->  -.  A. x  -.  x  e.  A
)
43con4bii 289 1  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   (/)c0 3620
This theorem is referenced by:  0el  3636  ssdif0  3678  difin0ss  3686  inssdif0  3687  ralf0  3726  disjiun  4194  0ex  4331  dm0  5075  reldm0  5079  uzwo  10531  uzwoOLD  10532  fzouzdisj  11161  hashgt0elex  11662  hausdiag  17669  rnelfmlem  17976  wzel  25567  nninfnub  26446  prtlem14  26714  stoweidlem34  27750  stoweidlem44  27760  bnj1476  29155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-nul 3621
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