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Theorem nntri3 6272
 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 4370 . . . . . 6 ¬ 𝐴𝐴
2 eleq2 2152 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 635 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 593 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
54adantl 272 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
6 simpl 108 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐴𝐵)
76con2i 593 . . . 4 (𝐴𝐵 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
87adantl 272 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
95, 82falsed 654 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
10 simpr 109 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11 eleq1 2151 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
121, 11mtbii 635 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
133, 12jca 301 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1413adantl 272 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1510, 142thd 174 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
1612con2i 593 . . . 4 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1716adantl 272 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ 𝐴 = 𝐵)
18 simpr 109 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐵𝐴)
1918con2i 593 . . . 4 (𝐵𝐴 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2019adantl 272 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2117, 202falsed 654 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
22 nntri3or 6268 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
239, 15, 21, 22mpjao3dan 1244 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290   ∈ wcel 1439  ωcom 4418 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416 This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-int 3695  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419 This theorem is referenced by:  frec2uzf1od  9867  nnti  12158
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