Theorem nn01to3 9576

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 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))

Proof of Theorem nn01to3

Step	Hyp	Ref	Expression
1		simp2 993	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 1 ≤ 𝑁)
2		simp1 992	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℕ0)
3		1z 9238	. . . . . . . . 9 ⊢ 1 ∈ ℤ
4		nn0z 9232	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
5		zleloe 9259	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
6	3, 4, 5	sylancr 412	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
7	2, 6	syl 14	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
8	1, 7	mpbid 146	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 < 𝑁 ∨ 1 = 𝑁))
9		1nn0 9151	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
10		nn0ltp1le 9274	. . . . . . . . . . 11 ⊢ ((1 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
11	9, 10	mpan 422	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
12		df-2 8937	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
13	12	breq1i 3996	. . . . . . . . . 10 ⊢ (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁)
14	11, 13	bitr4di 197	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ 2 ≤ 𝑁))
15		2z 9240	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
16		zleloe 9259	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
17	15, 4, 16	sylancr 412	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
18	14, 17	bitrd 187	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
19	18	orbi1d 786	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ((1 < 𝑁 ∨ 1 = 𝑁) ↔ ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁)))
20	2, 19	syl 14	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → ((1 < 𝑁 ∨ 1 = 𝑁) ↔ ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁)))
21	8, 20	mpbid 146	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁))
22	21	orcomd 724	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)))
23		orcom 723	. . . . 5 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) ↔ (2 = 𝑁 ∨ 2 < 𝑁))
24	23	orbi2i 757	. . . 4 ⊢ ((1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
25	22, 24	sylib 121	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
26		3orass 976	. . 3 ⊢ ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
27	25, 26	sylibr 133	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁))
28		3mix1 1161	. . . . 5 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
29	28	eqcoms 2173	. . . 4 ⊢ (1 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
30	29	a1i 9	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
31		3mix2 1162	. . . . 5 ⊢ (𝑁 = 2 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
32	31	eqcoms 2173	. . . 4 ⊢ (2 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
33	32	a1i 9	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
34		simp3 994	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ≤ 3)
35	34	biantrurd 303	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (3 ≤ 𝑁 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
36		2nn0 9152	. . . . . . . 8 ⊢ 2 ∈ ℕ0
37		nn0ltp1le 9274	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
38	36, 37	mpan 422	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
39		df-3 8938	. . . . . . . 8 ⊢ 3 = (2 + 1)
40	39	breq1i 3996	. . . . . . 7 ⊢ (3 ≤ 𝑁 ↔ (2 + 1) ≤ 𝑁)
41	38, 40	bitr4di 197	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ 3 ≤ 𝑁))
42	2, 41	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 ↔ 3 ≤ 𝑁))
43	2	nn0red 9189	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℝ)
44		3re 8952	. . . . . 6 ⊢ 3 ∈ ℝ
45		letri3 8000	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
46	43, 44, 45	sylancl 411	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
47	35, 42, 46	3bitr4d 219	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 ↔ 𝑁 = 3))
48		3mix3 1163	. . . 4 ⊢ (𝑁 = 3 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
49	47, 48	syl6bi 162	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
50	30, 33, 49	3jaod 1299	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
51	27, 50	mpd 13	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   ∨ w3o 972   ∧ w3a 973   = wceq 1348   ∈ wcel 2141   class class class wbr 3989  (class class class)co 5853  ℝcr 7773  1c1 7775   + caddc 7777   < clt 7954   ≤ cle 7955  2c2 8929  3c3 8930  ℕ₀cn0 9135  ℤcz 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-3 8938  df-n0 9136  df-z 9213

This theorem is referenced by: (None)