Theorem nn01to3 9606

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 998	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 1 ≤ 𝑁)
2		simp1 997	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℕ0)
3		1z 9268	. . . . . . . . 9 ⊢ 1 ∈ ℤ
4		nn0z 9262	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
5		zleloe 9289	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
6	3, 4, 5	sylancr 414	. . . . . . . 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 147	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 < 𝑁 ∨ 1 = 𝑁))
9		1nn0 9181	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
10		nn0ltp1le 9304	. . . . . . . . . . 11 ⊢ ((1 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
11	9, 10	mpan 424	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
12		df-2 8967	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
13	12	breq1i 4007	. . . . . . . . . 10 ⊢ (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁)
14	11, 13	bitr4di 198	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ 2 ≤ 𝑁))
15		2z 9270	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
16		zleloe 9289	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
17	15, 4, 16	sylancr 414	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
18	14, 17	bitrd 188	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
19	18	orbi1d 791	. . . . . . 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 147	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁))
22	21	orcomd 729	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)))
23		orcom 728	. . . . 5 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) ↔ (2 = 𝑁 ∨ 2 < 𝑁))
24	23	orbi2i 762	. . . 4 ⊢ ((1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
25	22, 24	sylib 122	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
26		3orass 981	. . 3 ⊢ ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
27	25, 26	sylibr 134	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁))
28		3mix1 1166	. . . . 5 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
29	28	eqcoms 2180	. . . 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 1167	. . . . 5 ⊢ (𝑁 = 2 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
32	31	eqcoms 2180	. . . 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 999	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ≤ 3)
35	34	biantrurd 305	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (3 ≤ 𝑁 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
36		2nn0 9182	. . . . . . . 8 ⊢ 2 ∈ ℕ0
37		nn0ltp1le 9304	. . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
38	36, 37	mpan 424	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
39		df-3 8968	. . . . . . . 8 ⊢ 3 = (2 + 1)
40	39	breq1i 4007	. . . . . . 7 ⊢ (3 ≤ 𝑁 ↔ (2 + 1) ≤ 𝑁)
41	38, 40	bitr4di 198	. . . . . 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 9219	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℝ)
44		3re 8982	. . . . . 6 ⊢ 3 ∈ ℝ
45		letri3 8028	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
46	43, 44, 45	sylancl 413	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
47	35, 42, 46	3bitr4d 220	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 ↔ 𝑁 = 3))
48		3mix3 1168	. . . 4 ⊢ (𝑁 = 3 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
49	47, 48	syl6bi 163	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
50	30, 33, 49	3jaod 1304	. 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 104   ↔ wb 105   ∨ wo 708   ∨ w3o 977   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   class class class wbr 4000  (class class class)co 5869  ℝcr 7801  1c1 7803   + caddc 7805   < clt 7982   ≤ cle 7983  2c2 8959  3c3 8960  ℕ₀cn0 9165  ℤcz 9242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-2 8967  df-3 8968  df-n0 9166  df-z 9243

This theorem is referenced by: (None)