Theorem 0npi 7408

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 2204	. 2 |- (/) = (/)
2		elni 7403	. . . 4 |- ( (/) e. N. <-> ( (/) e. om /\ (/) =/= (/) ) )
3	2	simprbi 275	. . 3 |- ( (/) e. N. -> (/) =/= (/) )
4	3	necon2bi 2430	. 2 |- ( (/) = (/) -> -. (/) e. N. )
5	1, 4	ax-mp 5	1 |- -. (/) e. N.

Syntax hints:   -. wn 3   = wceq 1372   e. wcel 2175   =/= wne 2375   (/)c0 3459   omcom 4636   N.cnpi 7367

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-v 2773  df-dif 3167  df-sn 3638  df-ni 7399

This theorem is referenced by:  elni2  7409