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Theorem 0ncn 11174
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 5718 . 2 ¬ ∅ ∈ (R × R)
2 df-c 11162 . . 3 ℂ = (R × R)
32eleq2i 2832 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 323 1 ¬ ∅ ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  c0 4332   × cxp 5682  Rcnr 10906  cc 11154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-c 11162
This theorem is referenced by:  axaddf  11186  axmulf  11187  bj-inftyexpitaudisj  37207  bj-inftyexpidisj  37212
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