Theorem 0ncn 7830

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 7831 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 4655	. 2 ⊢ ¬ ∅ ∈ (R × R)
2		df-c 7817	. . 3 ⊢ ℂ = (R × R)
3	2	eleq2i 2244	. 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
4	1, 3	mtbir 671	1 ⊢ ¬ ∅ ∈ ℂ

Syntax hints:  ¬ wn 3   ∈ wcel 2148  ∅c0 3423   × cxp 4625  Rcnr 7296  ℂcc 7809

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633  df-c 7817

This theorem is referenced by: (None)