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Theorem nntri3 6190
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
Assertion
Ref Expression
nntri3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))

Proof of Theorem nntri3
StepHypRef Expression
1 elirr 4320 . . . . . 6 ¬ 𝐴𝐴
2 eleq2 2146 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 632 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 590 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
54adantl 271 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
6 simpl 107 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐴𝐵)
76con2i 590 . . . 4 (𝐴𝐵 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
87adantl 271 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
95, 82falsed 651 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
10 simpr 108 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11 eleq1 2145 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
121, 11mtbii 632 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
133, 12jca 300 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1413adantl 271 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1510, 142thd 173 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
1612con2i 590 . . . 4 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1716adantl 271 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ 𝐴 = 𝐵)
18 simpr 108 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐵𝐴)
1918con2i 590 . . . 4 (𝐵𝐴 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2019adantl 271 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2117, 202falsed 651 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
22 nntri3or 6186 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
239, 15, 21, 22mpjao3dan 1239 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  ωcom 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-tr 3902  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369
This theorem is referenced by:  frec2uzf1od  9702
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