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Theorem 0ncn 7661
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. See also cnm 7662 which is a related property. (Contributed by NM, 2-May-1996.)
Assertion
Ref Expression
0ncn  |-  -.  (/)  e.  CC

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4573 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 7648 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2207 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 661 1  |-  -.  (/)  e.  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1481   (/)c0 3366    X. cxp 4543   R.cnr 7127   CCcc 7640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-opab 3996  df-xp 4551  df-c 7648
This theorem is referenced by: (None)
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