Theorem 0ncn 11086

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

Step	Hyp	Ref	Expression
1		0nelxp 5672	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 11074	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2820	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 323	1 ⊢ ¬ ∅ ∈ ℂ

Syntax hints:  ¬ wn 3   ∈ wcel 2109  ∅c0 4296   × cxp 5636  Rcnr 10818  ℂcc 11066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-xp 5644  df-c 11074

This theorem is referenced by:  axaddf  11098  axmulf  11099  bj-inftyexpitaudisj  37193  bj-inftyexpidisj  37198