Theorem nn01to3 11171

Description: A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)

Assertion

Ref	Expression
nn01to3	|- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )

Proof of Theorem nn01to3

Step	Hyp	Ref	Expression
1		3mix3 1167	. . 3 |- ( N = 3 -> ( N = 1 \/ N = 2 \/ N = 3 ) )
2	1	a1d 25	. 2 |- ( N = 3 -> ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) ) )
3		nn0re 10800	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1017	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N e. RR )
5		3re 10605	. . . . . . . . . 10 |- 3 e. RR
6	5	a1i 11	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> 3 e. RR )
7		simp3 998	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N <_ 3 )
8	4, 6, 7	leltned 9731	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 <-> 3 =/= N ) )
9		nesym 2739	. . . . . . . 8 |- ( 3 =/= N <-> -. N = 3 )
10	8, 9	syl6rbb 262	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 <-> N < 3 ) )
11		elnnnn0c 10837	. . . . . . . . 9 |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) )
12		orc 385	. . . . . . . . . . . 12 |- ( N = 1 -> ( N = 1 \/ N = 2 ) )
13	12	a1d 25	. . . . . . . . . . 11 |- ( N = 1 -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
14	13	a1d 25	. . . . . . . . . 10 |- ( N = 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
15		eluz2b3 11151	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
16		eluz2 11084	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
17		ax-1 6	. . . . . . . . . . . . . . . . . 18 |- ( N = 2 -> ( N < 3 -> N = 2 ) )
18	17	a1d 25	. . . . . . . . . . . . . . . . 17 |- ( N = 2 -> ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) ) )
19		zre 10864	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 e. ZZ -> 2 e. RR )
20		zre 10864	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> N e. RR )
21		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 <_ N -> 2 <_ N )
22		leltne 9670	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2 e. RR /\ N e. RR /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
23	19, 20, 21, 22	syl3an 1270	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
24		2z 10892	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 2 e. ZZ
25	24	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 e. ZZ )
26		simpr 461	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 < N )
27		df-3 10591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 3 = ( 2 + 1 )
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> 3 = ( 2 + 1 ) )
29	28	breq2d 4459	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( N < 3 <-> N < ( 2 + 1 ) ) )
30	29	biimpa 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ N < 3 ) -> N < ( 2 + 1 ) )
31	30	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> N < ( 2 + 1 ) )
32		btwnnz 10933	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( 2 e. ZZ /\ 2 < N /\ N < ( 2 + 1 ) ) -> -. N e. ZZ )
33	25, 26, 31, 32	syl3anc 1228	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> -. N e. ZZ )
34	33	pm2.21d 106	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> ( N e. ZZ -> N = 2 ) )
35	34	exp31 604	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ZZ -> ( N < 3 -> ( 2 < N -> ( N e. ZZ -> N = 2 ) ) ) )
36	35	com24 87	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ZZ -> ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) ) )
37	36	pm2.43i 47	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
38	37	3ad2ant2 1018	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
39	23, 38	sylbird 235	. . . . . . . . . . . . . . . . . 18 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N =/= 2 -> ( N < 3 -> N = 2 ) ) )
40	39	com12 31	. . . . . . . . . . . . . . . . 17 |- ( N =/= 2 -> ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) ) )
41	18, 40	pm2.61ine 2780	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) )
42	16, 41	sylbi 195	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> N = 2 ) )
43	42	imp 429	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> N = 2 )
44	43	olcd 393	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> ( N = 1 \/ N = 2 ) )
45	44	ex 434	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
46	15, 45	sylbir 213	. . . . . . . . . . 11 |- ( ( N e. NN /\ N =/= 1 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
47	46	expcom 435	. . . . . . . . . 10 |- ( N =/= 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
48	14, 47	pm2.61ine 2780	. . . . . . . . 9 |- ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
49	11, 48	sylbir 213	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
50	49	3adant3 1016	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
51	10, 50	sylbid 215	. . . . . 6 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 -> ( N = 1 \/ N = 2 ) ) )
52	51	impcom 430	. . . . 5 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 ) )
53	52	orcd 392	. . . 4 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
54		df-3or 974	. . . 4 |- ( ( N = 1 \/ N = 2 \/ N = 3 ) <-> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
55	53, 54	sylibr 212	. . 3 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
56	55	ex 434	. 2 |- ( -. N = 3 -> ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) ) )
57	2, 56	pm2.61i 164	1 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 972   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   1c1 9489   + caddc 9491   < clt 9624   <_ cle 9625   NNcn 10532   2c2 10581   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079

This theorem is referenced by:  hash1to3  12492