Theorem elni2 7146

Description: Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)

Assertion

Ref	Expression
elni2	|- ( A e. N. <-> ( A e. om /\ (/) e. A ) )

Proof of Theorem elni2

Step	Hyp	Ref	Expression
1		pinn 7141	. . 3 |- ( A e. N. -> A e. om )
2		0npi 7145	. . . . . 6 |- -. (/) e. N.
3		eleq1 2203	. . . . . 6 |- ( A = (/) -> ( A e. N. <-> (/) e. N. ) )
4	2, 3	mtbiri 665	. . . . 5 |- ( A = (/) -> -. A e. N. )
5	4	con2i 617	. . . 4 |- ( A e. N. -> -. A = (/) )
6		0elnn 4540	. . . . . 6 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
7	1, 6	syl 14	. . . . 5 |- ( A e. N. -> ( A = (/) \/ (/) e. A ) )
8	7	ord 714	. . . 4 |- ( A e. N. -> ( -. A = (/) -> (/) e. A ) )
9	5, 8	mpd 13	. . 3 |- ( A e. N. -> (/) e. A )
10	1, 9	jca 304	. 2 |- ( A e. N. -> ( A e. om /\ (/) e. A ) )
11		nndceq0 4539	. . . . . 6 |- ( A e. om -> DECID A = (/) )
12		df-dc 821	. . . . . 6 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
13	11, 12	sylib 121	. . . . 5 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
14	13	anim1i 338	. . . 4 |- ( ( A e. om /\ (/) e. A ) -> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
15		ancom 264	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
16		andi 808	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
17	15, 16	bitr3i 185	. . . 4 |- ( ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
18	14, 17	sylib 121	. . 3 |- ( ( A e. om /\ (/) e. A ) -> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
19		noel 3372	. . . . . . . . 9 |- -. (/) e. (/)
20		eleq2 2204	. . . . . . . . 9 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
21	19, 20	mtbiri 665	. . . . . . . 8 |- ( A = (/) -> -. (/) e. A )
22	21	pm2.21d 609	. . . . . . 7 |- ( A = (/) -> ( (/) e. A -> A e. N. ) )
23	22	impcom 124	. . . . . 6 |- ( ( (/) e. A /\ A = (/) ) -> A e. N. )
24	23	a1i 9	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ A = (/) ) -> A e. N. ) )
25		df-ne 2310	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
26		elni 7140	. . . . . . . 8 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
27	26	simplbi2 383	. . . . . . 7 |- ( A e. om -> ( A =/= (/) -> A e. N. ) )
28	25, 27	syl5bir 152	. . . . . 6 |- ( A e. om -> ( -. A = (/) -> A e. N. ) )
29	28	adantld 276	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ -. A = (/) ) -> A e. N. ) )
30	24, 29	jaod 707	. . . 4 |- ( A e. om -> ( ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) -> A e. N. ) )
31	30	adantr 274	. . 3 |- ( ( A e. om /\ (/) e. A ) -> ( ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) -> A e. N. ) )
32	18, 31	mpd 13	. 2 |- ( ( A e. om /\ (/) e. A ) -> A e. N. )
33	10, 32	impbii 125	1 |- ( A e. N. <-> ( A e. om /\ (/) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   = wceq 1332   e. wcel 1481   =/= wne 2309   (/)c0 3368   omcom 4512   N.cnpi 7104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513  df-ni 7136

This theorem is referenced by:  addclpi  7159  mulclpi  7160  mulcanpig  7167  addnidpig  7168  ltexpi  7169  ltmpig  7171  nnppipi  7175  archnqq  7249  enq0tr  7266