Theorem 0ncn 9954

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.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 0ncn

Step	Hyp	Ref	Expression
1		0nelxp 5143	. 2 |- -. (/) e. ( R. X. R. )
2		df-c 9942	. . 3 |- CC = ( R. X. R. )
3	2	eleq2i 2693	. 2 |- ( (/) e. CC <-> (/) e. ( R. X. R. ) )
4	1, 3	mtbir 313	1 |- -. (/) e. CC

Syntax hints:   -. wn 3   e. wcel 1990   (/)c0 3915   X. cxp 5112   R.cnr 9687   CCcc 9934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-c 9942

This theorem is referenced by:  axaddf  9966  axmulf  9967  bj-inftyexpidisj  33097