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Theorem 0ncn 10531
 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 5563 . 2 ¬ ∅ ∈ (R × R)
2 df-c 10519 . . 3 ℂ = (R × R)
32eleq2i 2902 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 325 1 ¬ ∅ ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∅c0 4267   × cxp 5527  Rcnr 10263  ℂcc 10511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-opab 5103  df-xp 5535  df-c 10519 This theorem is referenced by:  axaddf  10543  axmulf  10544  bj-inftyexpitaudisj  34501  bj-inftyexpidisj  34506
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