Theorem 0ncn 11257

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 5692	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 11245	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2856	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 326	1 ⊢ ¬ ∅ ∈ ℂ

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  ∅c0 4281   × cxp 5656  Rcnr 10989  ℂcc 11237

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174  df-xp 5664  df-c 11245

This theorem is used by:  axaddf  11269  axmulf  11270  bj-inftyexpitaudisj  38394  bj-inftyexpidisj  38399