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Theorem nntri3 6523
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 4558 . . . . . 6 ¬ 𝐴𝐴
2 eleq2 2253 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 675 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 628 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
54adantl 277 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
6 simpl 109 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐴𝐵)
76con2i 628 . . . 4 (𝐴𝐵 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
87adantl 277 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
95, 82falsed 703 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
10 simpr 110 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11 eleq1 2252 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
121, 11mtbii 675 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
133, 12jca 306 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1413adantl 277 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1510, 142thd 175 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
1612con2i 628 . . . 4 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1716adantl 277 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ 𝐴 = 𝐵)
18 simpr 110 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐵𝐴)
1918con2i 628 . . . 4 (𝐵𝐴 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2019adantl 277 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2117, 202falsed 703 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
22 nntri3or 6519 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
239, 15, 21, 22mpjao3dan 1318 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608
This theorem is referenced by:  frec2uzf1od  10439  nnti  15223
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