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Theorem 0ncn 11150
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 5706 . 2 ¬ ∅ ∈ (R × R)
2 df-c 11138 . . 3 ℂ = (R × R)
32eleq2i 2821 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 323 1 ¬ ∅ ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2099  c0 4318   × cxp 5670  Rcnr 10882  cc 11130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5205  df-xp 5678  df-c 11138
This theorem is referenced by:  axaddf  11162  axmulf  11163  bj-inftyexpitaudisj  36678  bj-inftyexpidisj  36683
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