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Theorem 0ncn 9833
 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 ¬ ∅ ∈ ℂ

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 5067 . 2 ¬ ∅ ∈ (R × R)
2 df-c 9821 . . 3 ℂ = (R × R)
32eleq2i 2680 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 312 1 ¬ ∅ ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1977  ∅c0 3874   × cxp 5036  Rcnr 9566  ℂcc 9813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-c 9821 This theorem is referenced by:  axaddf  9845  axmulf  9846  bj-inftyexpidisj  32274
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