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Theorem 0ncn 10543
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 5582 . 2 ¬ ∅ ∈ (R × R)
2 df-c 10531 . . 3 ℂ = (R × R)
32eleq2i 2901 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 324 1 ¬ ∅ ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2105  c0 4288   × cxp 5546  Rcnr 10275  cc 10523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-c 10531
This theorem is referenced by:  axaddf  10555  axmulf  10556  bj-inftyexpitaudisj  34379  bj-inftyexpidisj  34384
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