Theorem elni2 6412

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 6407	. . 3 |- ( A e. N. -> A e. om )
2		0npi 6411	. . . . . 6 |- -. (/) e. N.
3		eleq1 2100	. . . . . 6 |- ( A = (/) -> ( A e. N. <-> (/) e. N. ) )
4	2, 3	mtbiri 600	. . . . 5 |- ( A = (/) -> -. A e. N. )
5	4	con2i 557	. . . 4 |- ( A e. N. -> -. A = (/) )
6		0elnn 4340	. . . . . 6 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
7	1, 6	syl 14	. . . . 5 |- ( A e. N. -> ( A = (/) \/ (/) e. A ) )
8	7	ord 643	. . . 4 |- ( A e. N. -> ( -. A = (/) -> (/) e. A ) )
9	5, 8	mpd 13	. . 3 |- ( A e. N. -> (/) e. A )
10	1, 9	jca 290	. 2 |- ( A e. N. -> ( A e. om /\ (/) e. A ) )
11		nndceq0 4339	. . . . . 6 |- ( A e. om -> DECID A = (/) )
12		df-dc 743	. . . . . 6 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
13	11, 12	sylib 127	. . . . 5 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
14	13	anim1i 323	. . . 4 |- ( ( A e. om /\ (/) e. A ) -> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
15		ancom 253	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
16		andi 731	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
17	15, 16	bitr3i 175	. . . 4 |- ( ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
18	14, 17	sylib 127	. . 3 |- ( ( A e. om /\ (/) e. A ) -> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
19		noel 3228	. . . . . . . . 9 |- -. (/) e. (/)
20		eleq2 2101	. . . . . . . . 9 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
21	19, 20	mtbiri 600	. . . . . . . 8 |- ( A = (/) -> -. (/) e. A )
22	21	pm2.21d 549	. . . . . . 7 |- ( A = (/) -> ( (/) e. A -> A e. N. ) )
23	22	impcom 116	. . . . . 6 |- ( ( (/) e. A /\ A = (/) ) -> A e. N. )
24	23	a1i 9	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ A = (/) ) -> A e. N. ) )
25		df-ne 2206	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
26		elni 6406	. . . . . . . 8 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
27	26	simplbi2 367	. . . . . . 7 |- ( A e. om -> ( A =/= (/) -> A e. N. ) )
28	25, 27	syl5bir 142	. . . . . 6 |- ( A e. om -> ( -. A = (/) -> A e. N. ) )
29	28	adantld 263	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ -. A = (/) ) -> A e. N. ) )
30	24, 29	jaod 637	. . . 4 |- ( A e. om -> ( ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) -> A e. N. ) )
31	30	adantr 261	. . 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 117	1 |- ( A e. N. <-> ( A e. om /\ (/) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742   = wceq 1243   e. wcel 1393   =/= wne 2204   (/)c0 3224   omcom 4313   N.cnpi 6370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-ni 6402

This theorem is referenced by:  addclpi  6425  mulclpi  6426  mulcanpig  6433  addnidpig  6434  ltexpi  6435  ltmpig  6437  nnppipi  6441  archnqq  6515  enq0tr  6532