Theorem nn01to3 11781

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 1232	. . 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 11301	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1082	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N e. RR )
5		3re 11094	. . . . . . . . . 10 |- 3 e. RR
6	5	a1i 11	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> 3 e. RR )
7		simp3 1063	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N <_ 3 )
8	4, 6, 7	leltned 10190	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 <-> 3 =/= N ) )
9		nesym 2850	. . . . . . . 8 |- ( 3 =/= N <-> -. N = 3 )
10	8, 9	syl6rbb 277	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 <-> N < 3 ) )
11		elnnnn0c 11338	. . . . . . . . 9 |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) )
12		orc 400	. . . . . . . . . . . 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 11762	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
16		eluz2 11693	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
17		2a1 28	. . . . . . . . . . . . . . . . 17 |- ( N = 2 -> ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) ) )
18		zre 11381	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 e. ZZ -> 2 e. RR )
19		zre 11381	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> N e. RR )
20		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 <_ N -> 2 <_ N )
21		leltne 10127	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2 e. RR /\ N e. RR /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
22	18, 19, 20, 21	syl3an 1368	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
23		2z 11409	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 2 e. ZZ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 e. ZZ )
25		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 < N )
26		df-3 11080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 3 = ( 2 + 1 )
27	26	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> 3 = ( 2 + 1 ) )
28	27	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( N < 3 <-> N < ( 2 + 1 ) ) )
29	28	biimpa 501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ N < 3 ) -> N < ( 2 + 1 ) )
30	29	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> N < ( 2 + 1 ) )
31		btwnnz 11453	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( 2 e. ZZ /\ 2 < N /\ N < ( 2 + 1 ) ) -> -. N e. ZZ )
32	24, 25, 30, 31	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> -. N e. ZZ )
33	32	pm2.21d 118	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> ( N e. ZZ -> N = 2 ) )
34	33	exp31 630	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ZZ -> ( N < 3 -> ( 2 < N -> ( N e. ZZ -> N = 2 ) ) ) )
35	34	com24 95	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ZZ -> ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) ) )
36	35	pm2.43i 52	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
37	36	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
38	22, 37	sylbird 250	. . . . . . . . . . . . . . . . . 18 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N =/= 2 -> ( N < 3 -> N = 2 ) ) )
39	38	com12 32	. . . . . . . . . . . . . . . . 17 |- ( N =/= 2 -> ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) ) )
40	17, 39	pm2.61ine 2877	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) )
41	16, 40	sylbi 207	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> N = 2 ) )
42	41	imp 445	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> N = 2 )
43	42	olcd 408	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> ( N = 1 \/ N = 2 ) )
44	43	ex 450	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
45	15, 44	sylbir 225	. . . . . . . . . . 11 |- ( ( N e. NN /\ N =/= 1 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
46	45	expcom 451	. . . . . . . . . 10 |- ( N =/= 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
47	14, 46	pm2.61ine 2877	. . . . . . . . 9 |- ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
48	11, 47	sylbir 225	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
49	48	3adant3 1081	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
50	10, 49	sylbid 230	. . . . . 6 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 -> ( N = 1 \/ N = 2 ) ) )
51	50	impcom 446	. . . . 5 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 ) )
52	51	orcd 407	. . . 4 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
53		df-3or 1038	. . . 4 |- ( ( N = 1 \/ N = 2 \/ N = 3 ) <-> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
54	52, 53	sylibr 224	. . 3 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
55	54	ex 450	. 2 |- ( -. N = 3 -> ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) ) )
56	2, 55	pm2.61i 176	1 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  hash1to3  13273