Theorem 0ncn 7946

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 7947 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 4704	. 2 |- -. (/) e. ( R. X. R. )
2		df-c 7933	. . 3 |- CC = ( R. X. R. )
3	2	eleq2i 2272	. 2 |- ( (/) e. CC <-> (/) e. ( R. X. R. ) )
4	1, 3	mtbir 673	1 |- -. (/) e. CC

Syntax hints:   -. wn 3   e. wcel 2176   (/)c0 3460   X. cxp 4674   R.cnr 7412   CCcc 7925

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4107  df-xp 4682  df-c 7933

This theorem is referenced by: (None)