Theorem elni2 6504

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 6499	. . 3 |- ( A e. N. -> A e. om )
2		0npi 6503	. . . . . 6 |- -. (/) e. N.
3		eleq1 2141	. . . . . 6 |- ( A = (/) -> ( A e. N. <-> (/) e. N. ) )
4	2, 3	mtbiri 632	. . . . 5 |- ( A = (/) -> -. A e. N. )
5	4	con2i 589	. . . 4 |- ( A e. N. -> -. A = (/) )
6		0elnn 4358	. . . . . 6 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
7	1, 6	syl 14	. . . . 5 |- ( A e. N. -> ( A = (/) \/ (/) e. A ) )
8	7	ord 675	. . . 4 |- ( A e. N. -> ( -. A = (/) -> (/) e. A ) )
9	5, 8	mpd 13	. . 3 |- ( A e. N. -> (/) e. A )
10	1, 9	jca 300	. 2 |- ( A e. N. -> ( A e. om /\ (/) e. A ) )
11		nndceq0 4357	. . . . . 6 |- ( A e. om -> DECID A = (/) )
12		df-dc 776	. . . . . 6 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
13	11, 12	sylib 120	. . . . 5 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
14	13	anim1i 333	. . . 4 |- ( ( A e. om /\ (/) e. A ) -> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
15		ancom 262	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
16		andi 764	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
17	15, 16	bitr3i 184	. . . 4 |- ( ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
18	14, 17	sylib 120	. . 3 |- ( ( A e. om /\ (/) e. A ) -> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
19		noel 3255	. . . . . . . . 9 |- -. (/) e. (/)
20		eleq2 2142	. . . . . . . . 9 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
21	19, 20	mtbiri 632	. . . . . . . 8 |- ( A = (/) -> -. (/) e. A )
22	21	pm2.21d 581	. . . . . . 7 |- ( A = (/) -> ( (/) e. A -> A e. N. ) )
23	22	impcom 123	. . . . . 6 |- ( ( (/) e. A /\ A = (/) ) -> A e. N. )
24	23	a1i 9	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ A = (/) ) -> A e. N. ) )
25		df-ne 2246	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
26		elni 6498	. . . . . . . 8 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
27	26	simplbi2 377	. . . . . . 7 |- ( A e. om -> ( A =/= (/) -> A e. N. ) )
28	25, 27	syl5bir 151	. . . . . 6 |- ( A e. om -> ( -. A = (/) -> A e. N. ) )
29	28	adantld 272	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ -. A = (/) ) -> A e. N. ) )
30	24, 29	jaod 669	. . . 4 |- ( A e. om -> ( ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) -> A e. N. ) )
31	30	adantr 270	. . 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 124	1 |- ( A e. N. <-> ( A e. om /\ (/) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775   = wceq 1284   e. wcel 1433   =/= wne 2245   (/)c0 3251   omcom 4331   N.cnpi 6462

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-ni 6494

This theorem is referenced by:  addclpi  6517  mulclpi  6518  mulcanpig  6525  addnidpig  6526  ltexpi  6527  ltmpig  6529  nnppipi  6533  archnqq  6607  enq0tr  6624