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Theorem nntri3 6361
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 4426 . . . . . 6 ¬ 𝐴𝐴
2 eleq2 2181 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 648 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 601 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
54adantl 275 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
6 simpl 108 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐴𝐵)
76con2i 601 . . . 4 (𝐴𝐵 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
87adantl 275 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
95, 82falsed 676 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
10 simpr 109 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11 eleq1 2180 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
121, 11mtbii 648 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
133, 12jca 304 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1413adantl 275 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1510, 142thd 174 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
1612con2i 601 . . . 4 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1716adantl 275 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ 𝐴 = 𝐵)
18 simpr 109 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐵𝐴)
1918con2i 601 . . . 4 (𝐵𝐴 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2019adantl 275 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2117, 202falsed 676 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
22 nntri3or 6357 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
239, 15, 21, 22mpjao3dan 1270 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 1316  wcel 1465  ωcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475
This theorem is referenced by:  frec2uzf1od  10147  nnti  13118
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