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Theorem 0ncn 7651
 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 7652 which is a related property. (Contributed by NM, 2-May-1996.)
Assertion
Ref Expression
0ncn ¬ ∅ ∈ ℂ

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4567 . 2 ¬ ∅ ∈ (R × R)
2 df-c 7638 . . 3 ℂ = (R × R)
32eleq2i 2206 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 660 1 ¬ ∅ ∈ ℂ
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∈ wcel 1480  ∅c0 3363   × cxp 4537  Rcnr 7117  ℂcc 7630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-c 7638 This theorem is referenced by: (None)
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