Theorem nn01to3 11264

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 1180	. . 3 |- ( N = 3 -> ( N = 1 \/ N = 2 \/ N = 3 ) )
2	1	a1d 26	. 2 |- ( N = 3 -> ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) ) )
3		nn0re 10885	. . . . . . . . . 10 |- ( N e. NN0 -> N e. RR )
4	3	3ad2ant1 1030	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N e. RR )
5		3re 10690	. . . . . . . . . 10 |- 3 e. RR
6	5	a1i 11	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> 3 e. RR )
7		simp3 1011	. . . . . . . . 9 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> N <_ 3 )
8	4, 6, 7	leltned 9793	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 <-> 3 =/= N ) )
9		nesym 2682	. . . . . . . 8 |- ( 3 =/= N <-> -. N = 3 )
10	8, 9	syl6rbb 266	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 <-> N < 3 ) )
11		elnnnn0c 10922	. . . . . . . . 9 |- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) )
12		orc 387	. . . . . . . . . . . 12 |- ( N = 1 -> ( N = 1 \/ N = 2 ) )
13	12	a1d 26	. . . . . . . . . . 11 |- ( N = 1 -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
14	13	a1d 26	. . . . . . . . . 10 |- ( N = 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
15		eluz2b3 11239	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
16		eluz2 11172	. . . . . . . . . . . . . . . 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 10948	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 e. ZZ -> 2 e. RR )
19		zre 10948	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> N e. RR )
20		id 22	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 <_ N -> 2 <_ N )
21		leltne 9728	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2 e. RR /\ N e. RR /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
22	18, 19, 20, 21	syl3an 1311	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N <-> N =/= 2 ) )
23		2z 10976	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 2 e. ZZ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 e. ZZ )
25		simpr 463	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> 2 < N )
26		df-3 10676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 3 = ( 2 + 1 )
27	26	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> 3 = ( 2 + 1 ) )
28	27	breq2d 4417	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( N < 3 <-> N < ( 2 + 1 ) ) )
29	28	biimpa 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ N < 3 ) -> N < ( 2 + 1 ) )
30	29	adantr 467	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> N < ( 2 + 1 ) )
31		btwnnz 11019	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( 2 e. ZZ /\ 2 < N /\ N < ( 2 + 1 ) ) -> -. N e. ZZ )
32	24, 25, 30, 31	syl3anc 1269	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> -. N e. ZZ )
33	32	pm2.21d 110	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ N < 3 ) /\ 2 < N ) -> ( N e. ZZ -> N = 2 ) )
34	33	exp31 609	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ZZ -> ( N < 3 -> ( 2 < N -> ( N e. ZZ -> N = 2 ) ) ) )
35	34	com24 90	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ZZ -> ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) ) )
36	35	pm2.43i 49	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ZZ -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
37	36	3ad2ant2 1031	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( 2 < N -> ( N < 3 -> N = 2 ) ) )
38	22, 37	sylbird 239	. . . . . . . . . . . . . . . . . 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 2709	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) -> ( N < 3 -> N = 2 ) )
41	16, 40	sylbi 199	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> N = 2 ) )
42	41	imp 431	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> N = 2 )
43	42	olcd 395	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ N < 3 ) -> ( N = 1 \/ N = 2 ) )
44	43	ex 436	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
45	15, 44	sylbir 217	. . . . . . . . . . 11 |- ( ( N e. NN /\ N =/= 1 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
46	45	expcom 437	. . . . . . . . . 10 |- ( N =/= 1 -> ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) ) )
47	14, 46	pm2.61ine 2709	. . . . . . . . 9 |- ( N e. NN -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
48	11, 47	sylbir 217	. . . . . . . 8 |- ( ( N e. NN0 /\ 1 <_ N ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
49	48	3adant3 1029	. . . . . . 7 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N < 3 -> ( N = 1 \/ N = 2 ) ) )
50	10, 49	sylbid 219	. . . . . 6 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( -. N = 3 -> ( N = 1 \/ N = 2 ) ) )
51	50	impcom 432	. . . . 5 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 ) )
52	51	orcd 394	. . . 4 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
53		df-3or 987	. . . 4 |- ( ( N = 1 \/ N = 2 \/ N = 3 ) <-> ( ( N = 1 \/ N = 2 ) \/ N = 3 ) )
54	52, 53	sylibr 216	. . 3 |- ( ( -. N = 3 /\ ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
55	54	ex 436	. 2 |- ( -. N = 3 -> ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) ) )
56	2, 55	pm2.61i 168	1 |- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   \/ w3o 985   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   RRcr 9543   1c1 9545   + caddc 9547   < clt 9680   <_ cle 9681   NNcn 10616   2c2 10666   3c3 10667   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167

This theorem is referenced by:  hash1to3  12655