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Theorem 0ncn 11125
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 5701 . 2 ¬ ∅ ∈ (R × R)
2 df-c 11113 . . 3 ℂ = (R × R)
32eleq2i 2817 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 323 1 ¬ ∅ ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  c0 4315   × cxp 5665  Rcnr 10857  cc 11105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202  df-xp 5673  df-c 11113
This theorem is referenced by:  axaddf  11137  axmulf  11138  bj-inftyexpitaudisj  36577  bj-inftyexpidisj  36582
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