Theorem 0npi 9704

Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
0npi	|- -. (/) e. N.

Proof of Theorem 0npi

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- (/) = (/)
2		elni 9698	. . . 4 |- ( (/) e. N. <-> ( (/) e. om /\ (/) =/= (/) ) )
3	2	simprbi 480	. . 3 |- ( (/) e. N. -> (/) =/= (/) )
4	3	necon2bi 2824	. 2 |- ( (/) = (/) -> -. (/) e. N. )
5	1, 4	ax-mp 5	1 |- -. (/) e. N.

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   omcom 7065   N.cnpi 9666

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-sn 4178  df-ni 9694

This theorem is referenced by:  addasspi  9717  mulasspi  9719  distrpi  9720  addcanpi  9721  mulcanpi  9722  addnidpi  9723  ltapi  9725  ltmpi  9726  ordpipq  9764