Theorem 0npi 7326

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 2187	. 2 |- (/) = (/)
2		elni 7321	. . . 4 |- ( (/) e. N. <-> ( (/) e. om /\ (/) =/= (/) ) )
3	2	simprbi 275	. . 3 |- ( (/) e. N. -> (/) =/= (/) )
4	3	necon2bi 2412	. 2 |- ( (/) = (/) -> -. (/) e. N. )
5	1, 4	ax-mp 5	1 |- -. (/) e. N.

Syntax hints:   -. wn 3   = wceq 1363   e. wcel 2158   =/= wne 2357   (/)c0 3434   omcom 4601   N.cnpi 7285

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-sn 3610  df-ni 7317

This theorem is referenced by:  elni2  7327