Theorem elni2 7255

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 7250	. . 3 |- ( A e. N. -> A e. om )
2		0npi 7254	. . . . . 6 |- -. (/) e. N.
3		eleq1 2229	. . . . . 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 4596	. . . . . 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 4595	. . . . . 6 |- ( A e. om -> DECID A = (/) )
12		df-dc 825	. . . . . 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 3413	. . . . . . . . 9 |- -. (/) e. (/)
20		eleq2 2230	. . . . . . . . 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 2337	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
26		elni 7249	. . . . . . . 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 824   = wceq 1343   e. wcel 2136   =/= wne 2336   (/)c0 3409   omcom 4567   N.cnpi 7213

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568  df-ni 7245

This theorem is referenced by:  addclpi  7268  mulclpi  7269  mulcanpig  7276  addnidpig  7277  ltexpi  7278  ltmpig  7280  nnppipi  7284  archnqq  7358  enq0tr  7375