Theorem 0ncn 5223

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.

Assertion

Ref	Expression
0ncn	|- -. (/) e. CC

Proof of Theorem 0ncn

Step	Hyp	Ref	Expression
1		0nelxp 3230	. 2 |- -. (/) e. (R. X. R.)
2		df-c 5212	. . 3 |- CC = (R. X. R.)
3	2	eleq2i 1530	. 2 |- ((/) e. CC <-> (/) e. (R. X. R.))
4	1, 3	mtbir 192	1 |- -. (/) e. CC

Syntax hints:  -. wn 2   e. wcel 955  (/)c0 2270   X. cxp 3158  R.cnr 4965  CCcc 5204

This theorem is referenced by:  axaddopr 5237  axmulopr 5238

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-c 5212

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