Theorem 0ncn 6403

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 4066	. 2 |- -. (/) e. (R. X. R.)
2		df-c 6392	. . 3 |- CC = (R. X. R.)
3	2	eleq2i 1961	. 2 |- ((/) e. CC <-> (/) e. (R. X. R.))
4	1, 3	mtbir 209	1 |- -. (/) e. CC

Syntax hints:  -. wn 2   e. wcel 1300  (/)c0 2875   X. cxp 3984  R.cnr 6145  CCcc 6384

This theorem is referenced by:  axaddopr 6417  axmulopr 6418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-opab 3396  df-xp 4000  df-c 6392

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