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Theorem pitri3or 6563
 Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
Assertion
Ref Expression
pitri3or ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))

Proof of Theorem pitri3or
StepHypRef Expression
1 pinn 6550 . . 3 (𝐴N𝐴 ∈ ω)
2 pinn 6550 . . 3 (𝐵N𝐵 ∈ ω)
3 nntri3or 6130 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
41, 2, 3syl2an 283 . 2 ((𝐴N𝐵N) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
5 ltpiord 6560 . . 3 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴𝐵))
6 biidd 170 . . 3 ((𝐴N𝐵N) → (𝐴 = 𝐵𝐴 = 𝐵))
7 ltpiord 6560 . . . 4 ((𝐵N𝐴N) → (𝐵 <N 𝐴𝐵𝐴))
87ancoms 264 . . 3 ((𝐴N𝐵N) → (𝐵 <N 𝐴𝐵𝐴))
95, 6, 83orbi123d 1243 . 2 ((𝐴N𝐵N) → ((𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴) ↔ (𝐴𝐵𝐴 = 𝐵𝐵𝐴)))
104, 9mpbird 165 1 ((𝐴N𝐵N) → (𝐴 <N 𝐵𝐴 = 𝐵𝐵 <N 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ w3o 919   = wceq 1285   ∈ wcel 1434   class class class wbr 3787  ωcom 4333  Ncnpi 6513
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