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Theorem 0ncn 9582
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 4880 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 9570 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2531 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 305 1  |-  -.  (/)  e.  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1897   (/)c0 3742    X. cxp 4850   R.cnr 9315   CCcc 9562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-opab 4475  df-xp 4858  df-c 9570
This theorem is referenced by:  axaddf  9594  axmulf  9595  bj-inftyexpidisj  31696
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