Theorem 0ncn 10146

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 5300	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 10134	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2831	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 312	1 ⊢ ¬ ∅ ∈ ℂ

Syntax hints:  ¬ wn 3   ∈ wcel 2139  ∅c0 4058   × cxp 5264  Rcnr 9879  ℂcc 10126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-c 10134

This theorem is referenced by:  axaddf  10158  axmulf  10159  bj-inftyexpidisj  33408