Theorem 0ncn 7979

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. See also cnm 7980 which is a related property. (Contributed by NM, 2-May-1996.)

Assertion

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

Proof of Theorem 0ncn

Step	Hyp	Ref	Expression
1		0nelxp 4721	. 2 |- -. (/) e. ( R. X. R. )
2		df-c 7966	. . 3 |- CC = ( R. X. R. )
3	2	eleq2i 2274	. 2 |- ( (/) e. CC <-> (/) e. ( R. X. R. ) )
4	1, 3	mtbir 673	1 |- -. (/) e. CC

Syntax hints:   -. wn 3   e. wcel 2178   (/)c0 3468   X. cxp 4691   R.cnr 7445   CCcc 7958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-xp 4699  df-c 7966

This theorem is referenced by: (None)