Theorem 0ncn 11091

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 5681	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 11079	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2854	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 325	1 ⊢ ¬ ∅ ∈ ℂ

Syntax hints:  ¬ wn 3   ∈ wcel 2142  ∅c0 4285   × cxp 5645  Rcnr 10823  ℂcc 11071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-c 11079

This theorem is referenced by:  axaddf  11103  axmulf  11104  bj-inftyexpitaudisj  37697  bj-inftyexpidisj  37702