Theorem 0ncn 8146

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. See also cnm 8147 which is a related property. (Contributed by NM, 2-May-1996.)

Assertion

Ref	Expression
0ncn	⊢ ¬ ∅ ∈ ℂ

Proof of Theorem 0ncn

Step	Hyp	Ref	Expression
1		0nelxp 4777	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 8133	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2299	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 678	1 ⊢ ¬ ∅ ∈ ℂ

Syntax hints:  ¬ wn 3   ∈ wcel 2203  ∅c0 3508   × cxp 4747  Rcnr 7612  ℂcc 8125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-xp 4755  df-c 8133

This theorem is referenced by: (None)