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Theorem nn01to3 9421
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 982 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 1 ≤ 𝑁)
2 simp1 981 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ∈ ℕ0)
3 1z 9092 . . . . . . . . 9 1 ∈ ℤ
4 nn0z 9086 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
5 zleloe 9113 . . . . . . . . 9 ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ 𝑁 ↔ (1 < 𝑁 ∨ 1 = 𝑁)))
63, 4, 5sylancr 410 . . . . . . . 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 146 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 < 𝑁 ∨ 1 = 𝑁))
9 1nn0 9005 . . . . . . . . . . 11 1 ∈ ℕ0
10 nn0ltp1le 9128 . . . . . . . . . . 11 ((1 ∈ ℕ0𝑁 ∈ ℕ0) → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
119, 10mpan 420 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁))
12 df-2 8791 . . . . . . . . . . 11 2 = (1 + 1)
1312breq1i 3936 . . . . . . . . . 10 (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁)
1411, 13syl6bbr 197 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ 2 ≤ 𝑁))
15 2z 9094 . . . . . . . . . 10 2 ∈ ℤ
16 zleloe 9113 . . . . . . . . . 10 ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1715, 4, 16sylancr 410 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1814, 17bitrd 187 . . . . . . . 8 (𝑁 ∈ ℕ0 → (1 < 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁)))
1918orbi1d 780 . . . . . . 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 146 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → ((2 < 𝑁 ∨ 2 = 𝑁) ∨ 1 = 𝑁))
2221orcomd 718 . . . 4 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)))
23 orcom 717 . . . . 5 ((2 < 𝑁 ∨ 2 = 𝑁) ↔ (2 = 𝑁 ∨ 2 < 𝑁))
2423orbi2i 751 . . . 4 ((1 = 𝑁 ∨ (2 < 𝑁 ∨ 2 = 𝑁)) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
2522, 24sylib 121 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
26 3orass 965 . . 3 ((1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁) ↔ (1 = 𝑁 ∨ (2 = 𝑁 ∨ 2 < 𝑁)))
2725, 26sylibr 133 . 2 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (1 = 𝑁 ∨ 2 = 𝑁 ∨ 2 < 𝑁))
28 3mix1 1150 . . . . 5 (𝑁 = 1 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
2928eqcoms 2142 . . . 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 1151 . . . . 5 (𝑁 = 2 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
3231eqcoms 2142 . . . 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 983 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ≤ 3)
3534biantrurd 303 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (3 ≤ 𝑁 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
36 2nn0 9006 . . . . . . . 8 2 ∈ ℕ0
37 nn0ltp1le 9128 . . . . . . . 8 ((2 ∈ ℕ0𝑁 ∈ ℕ0) → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
3836, 37mpan 420 . . . . . . 7 (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ (2 + 1) ≤ 𝑁))
39 df-3 8792 . . . . . . . 8 3 = (2 + 1)
4039breq1i 3936 . . . . . . 7 (3 ≤ 𝑁 ↔ (2 + 1) ≤ 𝑁)
4138, 40syl6bbr 197 . . . . . 6 (𝑁 ∈ ℕ0 → (2 < 𝑁 ↔ 3 ≤ 𝑁))
422, 41syl 14 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁 ↔ 3 ≤ 𝑁))
432nn0red 9043 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → 𝑁 ∈ ℝ)
44 3re 8806 . . . . . 6 3 ∈ ℝ
45 letri3 7857 . . . . . 6 ((𝑁 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
4643, 44, 45sylancl 409 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (𝑁 = 3 ↔ (𝑁 ≤ 3 ∧ 3 ≤ 𝑁)))
4735, 42, 463bitr4d 219 . . . 4 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁𝑁 = 3))
48 3mix3 1152 . . . 4 (𝑁 = 3 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
4947, 48syl6bi 162 . . 3 ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (2 < 𝑁 → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)))
5030, 33, 493jaod 1282 . 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 103  wb 104  wo 697  w3o 961  w3a 962   = wceq 1331  wcel 1480   class class class wbr 3929  (class class class)co 5774  cr 7631  1c1 7633   + caddc 7635   < clt 7812  cle 7813  2c2 8783  3c3 8784  0cn0 8989  cz 9066
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-2 8791  df-3 8792  df-n0 8990  df-z 9067
This theorem is referenced by: (None)
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