Theorem elni2 7374

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 7369	. . 3 |- ( A e. N. -> A e. om )
2		0npi 7373	. . . . . 6 |- -. (/) e. N.
3		eleq1 2256	. . . . . 6 |- ( A = (/) -> ( A e. N. <-> (/) e. N. ) )
4	2, 3	mtbiri 676	. . . . 5 |- ( A = (/) -> -. A e. N. )
5	4	con2i 628	. . . 4 |- ( A e. N. -> -. A = (/) )
6		0elnn 4651	. . . . . 6 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )
7	1, 6	syl 14	. . . . 5 |- ( A e. N. -> ( A = (/) \/ (/) e. A ) )
8	7	ord 725	. . . 4 |- ( A e. N. -> ( -. A = (/) -> (/) e. A ) )
9	5, 8	mpd 13	. . 3 |- ( A e. N. -> (/) e. A )
10	1, 9	jca 306	. 2 |- ( A e. N. -> ( A e. om /\ (/) e. A ) )
11		nndceq0 4650	. . . . . 6 |- ( A e. om -> DECID A = (/) )
12		df-dc 836	. . . . . 6 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
13	11, 12	sylib 122	. . . . 5 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
14	13	anim1i 340	. . . 4 |- ( ( A e. om /\ (/) e. A ) -> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
15		ancom 266	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) )
16		andi 819	. . . . 5 |- ( ( (/) e. A /\ ( A = (/) \/ -. A = (/) ) ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
17	15, 16	bitr3i 186	. . . 4 |- ( ( ( A = (/) \/ -. A = (/) ) /\ (/) e. A ) <-> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
18	14, 17	sylib 122	. . 3 |- ( ( A e. om /\ (/) e. A ) -> ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) )
19		noel 3450	. . . . . . . . 9 |- -. (/) e. (/)
20		eleq2 2257	. . . . . . . . 9 |- ( A = (/) -> ( (/) e. A <-> (/) e. (/) ) )
21	19, 20	mtbiri 676	. . . . . . . 8 |- ( A = (/) -> -. (/) e. A )
22	21	pm2.21d 620	. . . . . . 7 |- ( A = (/) -> ( (/) e. A -> A e. N. ) )
23	22	impcom 125	. . . . . 6 |- ( ( (/) e. A /\ A = (/) ) -> A e. N. )
24	23	a1i 9	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ A = (/) ) -> A e. N. ) )
25		df-ne 2365	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
26		elni 7368	. . . . . . . 8 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
27	26	simplbi2 385	. . . . . . 7 |- ( A e. om -> ( A =/= (/) -> A e. N. ) )
28	25, 27	biimtrrid 153	. . . . . 6 |- ( A e. om -> ( -. A = (/) -> A e. N. ) )
29	28	adantld 278	. . . . 5 |- ( A e. om -> ( ( (/) e. A /\ -. A = (/) ) -> A e. N. ) )
30	24, 29	jaod 718	. . . 4 |- ( A e. om -> ( ( ( (/) e. A /\ A = (/) ) \/ ( (/) e. A /\ -. A = (/) ) ) -> A e. N. ) )
31	30	adantr 276	. . 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 126	1 |- ( A e. N. <-> ( A e. om /\ (/) e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   (/)c0 3446   omcom 4622   N.cnpi 7332

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623  df-ni 7364

This theorem is referenced by:  addclpi  7387  mulclpi  7388  mulcanpig  7395  addnidpig  7396  ltexpi  7397  ltmpig  7399  nnppipi  7403  archnqq  7477  enq0tr  7494