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Theorem 0ncn 9558
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 4878 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 9546 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2500 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 300 1  |-  -.  (/)  e.  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1868   (/)c0 3761    X. cxp 4848   R.cnr 9291   CCcc 9538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-opab 4480  df-xp 4856  df-c 9546
This theorem is referenced by:  axaddf  9570  axmulf  9571  bj-inftyexpidisj  31604
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