Theorem nn01to3 11176

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 1165	. . 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 1015	. . . . . . . . 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 996	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N <_ 3 )
8	4, 6, 7	leltned 9725	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 <-> 3 =/= N ) )
9		nesym 2726	. . . . . . . 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 383	. . . . . . . . . . . 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 11156	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
16		eluz2 11088	. . . . . . . . . . . . . . . 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 9663	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2 e. RR /\ N e. RR /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
23	19, 20, 21, 22	syl3an 1268	. . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . . . . . 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 4451	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( N < 3 <-> N < ( 2 + 1 ) ) )
30	29	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ N < 3 ) -> N < ( 2 + 1 ) )
31	30	adantr 463	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> N < ( 2 + 1 ) )
32		btwnnz 10935	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( 2 e. ZZ /\ 2 < N /\ N < ( 2 + 1 ) ) -> -. N e. ZZ )
33	25, 26, 31, 32	syl3anc 1226	. . . . . . . . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> N = 2 )
44	43	olcd 391	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> ( N = 1 \/ N = 2 ) )
45	44	ex 432	. . . . . . . . . . . 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 433	. . . . . . . . . 10 |- ( N =/= 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
48	14, 47	pm2.61ine 2767	. . . . . . . . 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 1014	. . . . . . 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 428	. . . . 5 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 ) )
53	52	orcd 390	. . . 4 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
54		df-3or 972	. . . 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 432	. 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 366   /\ wa 367   \/ w3o 970   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482   + caddc 9484   < clt 9617   <_ cle 9618   NNcn 10531   2c2 10581   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083

This theorem is referenced by:  hash1to3  12514