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Theorem 0ncn 9992
 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 5177 . 2 ¬ ∅ ∈ (R × R)
2 df-c 9980 . . 3 ℂ = (R × R)
32eleq2i 2722 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 312 1 ¬ ∅ ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2030  ∅c0 3948   × cxp 5141  Rcnr 9725  ℂcc 9972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-c 9980 This theorem is referenced by:  axaddf  10004  axmulf  10005  bj-inftyexpidisj  33227
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