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Theorem 0ncn 9499
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 5016 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 9487 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2532 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 297 1  |-  -.  (/)  e.  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1823   (/)c0 3783    X. cxp 4986   R.cnr 9232   CCcc 9479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-c 9487
This theorem is referenced by:  axaddf  9511  axmulf  9512  bj-inftyexpidisj  35032
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