Theorem 0npi 7593

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

Assertion

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

Proof of Theorem 0npi

Step	Hyp	Ref	Expression
1		eqid 2231	. 2 |- (/) = (/)
2		elni 7588	. . . 4 |- ( (/) e. N. <-> ( (/) e. om /\ (/) =/= (/) ) )
3	2	simprbi 275	. . 3 |- ( (/) e. N. -> (/) =/= (/) )
4	3	necon2bi 2458	. 2 |- ( (/) = (/) -> -. (/) e. N. )
5	1, 4	ax-mp 5	1 |- -. (/) e. N.

Syntax hints:   -. wn 3   = wceq 1398   e. wcel 2202   =/= wne 2403   (/)c0 3496   omcom 4694   N.cnpi 7552

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-sn 3679  df-ni 7584

This theorem is referenced by:  elni2  7594