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Theorem 0ncn 11076
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 5672 . 2 ¬ ∅ ∈ (R × R)
2 df-c 11064 . . 3 ℂ = (R × R)
32eleq2i 2830 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 323 1 ¬ ∅ ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  c0 4287   × cxp 5636  Rcnr 10808  cc 11056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644  df-c 11064
This theorem is referenced by:  axaddf  11088  axmulf  11089  bj-inftyexpitaudisj  35705  bj-inftyexpidisj  35710
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