Theorem nn01to3 9808

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 1022	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 1 ≤ 𝑁)
2		simp1 1021	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℕ0)
3		1z 9468	. . . . . . . . 9 ⊢ 1 ∈ ℤ
4		nn0z 9462	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
5		zleloe 9489	. . . . . . . . 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 9381	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
10		nn0ltp1le 9505	. . . . . . . . . . 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 9165	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
13	12	breq1i 4089	. . . . . . . . . 10 ⊢ (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁)
14	11, 13	bitr4di 198	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ 2 ≤ 𝑁))
15		2z 9470	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
16		zleloe 9489	. . . . . . . . . 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 796	. . . . . . 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 734	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)))
23		orcom 733	. . . . 5 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) ↔ (2 = 𝑁 ∨ 2 < 𝑁))
24	23	orbi2i 767	. . . 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 1005	. . 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 1190	. . . . 5 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
29	28	eqcoms 2232	. . . 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 1191	. . . . 5 ⊢ (𝑁 = 2 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
32	31	eqcoms 2232	. . . 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 1023	. . . . . 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 9382	. . . . . . . 8 ⊢ 2 ∈ ℕ0
37		nn0ltp1le 9505	. . . . . . . 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 9166	. . . . . . . 8 ⊢ 3 = (2 + 1)
40	39	breq1i 4089	. . . . . . 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 9419	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → 𝑁 ∈ ℝ)
44		3re 9180	. . . . . 6 ⊢ 3 ∈ ℝ
45		letri3 8223	. . . . . 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 1192	. . . 4 ⊢ (𝑁 = 3 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
49	47, 48	biimtrdi 163	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (2 < 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
50	30, 33, 49	3jaod 1338	. 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 713   ∨ w3o 1001   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   class class class wbr 4082  (class class class)co 6000  ℝcr 7994  1c1 7996   + caddc 7998   < clt 8177   ≤ cle 8178  2c2 9157  3c3 9158  ℕ₀cn0 9365  ℤcz 9442

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-3 9166  df-n0 9366  df-z 9443

This theorem is referenced by: (None)