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Theorem 0ncn 7270
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.)
Assertion
Ref Expression
0ncn  |-  -.  (/)  e.  CC

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4426 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 7257 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2149 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 629 1  |-  -.  (/)  e.  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1434   (/)c0 3269    X. cxp 4397   R.cnr 6757   CCcc 7249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4405  df-c 7257
This theorem is referenced by: (None)
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