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Theorem pitric 6298
 Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
Assertion
Ref Expression
pitric ((A N B N) → (A <N B ↔ ¬ (A = B B <N A)))

Proof of Theorem pitric
StepHypRef Expression
1 pinn 6286 . . 3 (A NA 𝜔)
2 pinn 6286 . . 3 (B NB 𝜔)
3 nntri2 6005 . . 3 ((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
41, 2, 3syl2an 273 . 2 ((A N B N) → (A B ↔ ¬ (A = B B A)))
5 ltpiord 6296 . 2 ((A N B N) → (A <N BA B))
6 ltpiord 6296 . . . . 5 ((B N A N) → (B <N AB A))
76ancoms 255 . . . 4 ((A N B N) → (B <N AB A))
87orbi2d 703 . . 3 ((A N B N) → ((A = B B <N A) ↔ (A = B B A)))
98notbid 591 . 2 ((A N B N) → (¬ (A = B B <N A) ↔ ¬ (A = B B A)))
104, 5, 93bitr4d 209 1 ((A N B N) → (A <N B ↔ ¬ (A = B B <N A)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390   class class class wbr 3754  𝜔com 4255  Ncnpi 6249
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