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Theorem 0ncn 6966
 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.)
Assertion
Ref Expression
0ncn ¬ ∅ ∈ ℂ

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4400 . 2 ¬ ∅ ∈ (R × R)
2 df-c 6953 . . 3 ℂ = (R × R)
32eleq2i 2120 . 2 (∅ ∈ ℂ ↔ ∅ ∈ (R × R))
41, 3mtbir 606 1 ¬ ∅ ∈ ℂ
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∈ wcel 1409  ∅c0 3252   × cxp 4371  Rcnr 6453  ℂcc 6945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-c 6953 This theorem is referenced by: (None)
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