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Theorem 0ncn 7348
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  |-  -.  (/)  e.  CC

Proof of Theorem 0ncn
StepHypRef Expression
1 0nelxp 4455 . 2  |-  -.  (/)  e.  ( R.  X.  R. )
2 df-c 7335 . . 3  |-  CC  =  ( R.  X.  R. )
32eleq2i 2154 . 2  |-  ( (/)  e.  CC  <->  (/)  e.  ( R. 
X.  R. ) )
41, 3mtbir 631 1  |-  -.  (/)  e.  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1438   (/)c0 3284    X. cxp 4426   R.cnr 6835   CCcc 7327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-c 7335
This theorem is referenced by: (None)
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