Theorem 0ncn 11133

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 5697	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 11121	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2857	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 326	1 ⊢ ¬ ∅ ∈ ℂ

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  ∅c0 4286   × cxp 5661  Rcnr 10865  ℂcc 11113

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669  df-c 11121

This theorem is used by:  axaddf  11145  axmulf  11146  bj-inftyexpitaudisj  37906  bj-inftyexpidisj  37911