ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn01to3 GIF version

Theorem nn01to3 9616
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 ≤ 𝑁𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))

Proof of Theorem nn01to3
StepHypRef Expression
1 simp2 998 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 1 ≤ 𝑁)
2 simp1 997 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ∈ ℕ0)
3 1z 9278 . . . . . . . . 9 1 ∈ ℤ
4 nn0z 9272 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
5 zleloe 9299 . . . . . . . . 9 ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
63, 4, 5sylancr 414 . . . . . . . 8 (𝑁 ∈ ℕ0 → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
72, 6syl 14 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
81, 7mpbid 147 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 < 𝑁 ∨ 1 = 𝑁))
9 1nn0 9191 . . . . . . . . . . 11 1 ∈ ℕ0
10 nn0ltp1le 9314 . . . . . . . . . . 11 ((1 ∈ ℕ0𝑁 ∈ ℕ0) → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
119, 10mpan 424 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
12 df-2 8977 . . . . . . . . . . 11 2 = (1 + 1)
1312breq1i 4010 . . . . . . . . . 10 (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁)
1411, 13bitr4di 198 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ 2 ≤ 𝑁))
15 2z 9280 . . . . . . . . . 10 2 ∈ ℤ
16 zleloe 9299 . . . . . . . . . 10 ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1715, 4, 16sylancr 414 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1814, 17bitrd 188 . . . . . . . 8 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1918orbi1d 791 . . . . . . 7 (𝑁 ∈ ℕ0 → ((1 < 𝑁 ∨ 1 = 𝑁) ↔ ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁)))
202, 19syl 14 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → ((1 < 𝑁 ∨ 1 = 𝑁) ↔ ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁)))
218, 20mpbid 147 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁))
2221orcomd 729 . . . 4 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)))
23 orcom 728 . . . . 5 ((2 < 𝑁 ∨ 2 = 𝑁) ↔ (2 = 𝑁 ∨ 2 < 𝑁))
2423orbi2i 762 . . . 4 ((1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
2522, 24sylib 122 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
26 3orass 981 . . 3 ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
2725, 26sylibr 134 . 2 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁))
28 3mix1 1166 . . . . 5 (𝑁 = 1 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
2928eqcoms 2180 . . . 4 (1 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
3029a1i 9 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
31 3mix2 1167 . . . . 5 (𝑁 = 2 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
3231eqcoms 2180 . . . 4 (2 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
3332a1i 9 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 = 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
34 simp3 999 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ≤ 3)
3534biantrurd 305 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (3 ≤ 𝑁 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
36 2nn0 9192 . . . . . . . 8 2 ∈ ℕ0
37 nn0ltp1le 9314 . . . . . . . 8 ((2 ∈ ℕ0𝑁 ∈ ℕ0) → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
3836, 37mpan 424 . . . . . . 7 (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
39 df-3 8978 . . . . . . . 8 3 = (2 + 1)
4039breq1i 4010 . . . . . . 7 (3 ≤ 𝑁 ↔ (2 + 1) ≤ 𝑁)
4138, 40bitr4di 198 . . . . . 6 (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ 3 ≤ 𝑁))
422, 41syl 14 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁 ↔ 3 ≤ 𝑁))
432nn0red 9229 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ∈ ℝ)
44 3re 8992 . . . . . 6 3 ∈ ℝ
45 letri3 8037 . . . . . 6 ((𝑁 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
4643, 44, 45sylancl 413 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
4735, 42, 463bitr4d 220 . . . 4 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁𝑁 = 3))
48 3mix3 1168 . . . 4 (𝑁 = 3 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
4947, 48syl6bi 163 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
5030, 33, 493jaod 1304 . 2 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
5127, 50mpd 13 1 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3o 977  w3a 978   = wceq 1353  wcel 2148   class class class wbr 4003  (class class class)co 5874  cr 7809  1c1 7811   + caddc 7813   < clt 7991  cle 7992  2c2 8969  3c3 8970  0cn0 9175  cz 9252
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-2 8977  df-3 8978  df-n0 9176  df-z 9253
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator