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Theorem 0ncn 8755
Description: The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
0ncn  |-  -.  (/)  e.  CC

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4717 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 8743 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2347 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 290 1  |-  -.  (/)  e.  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684   (/)c0 3455    X. cxp 4687   R.cnr 8489   CCcc 8735
This theorem is referenced by:  axaddf  8767  axmulf  8768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-c 8743
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