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Theorem nntri3 6105
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 4293 . . . . . 6 ¬ 𝐴𝐴
2 eleq2 2117 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 609 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 567 . . . 4 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
54adantl 266 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
6 simpl 106 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐴𝐵)
76con2i 567 . . . 4 (𝐴𝐵 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
87adantl 266 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
95, 82falsed 628 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
10 simpr 107 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
11 eleq1 2116 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
121, 11mtbii 609 . . . . 5 (𝐴 = 𝐵 → ¬ 𝐵𝐴)
133, 12jca 294 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1413adantl 266 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
1510, 142thd 168 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 = 𝐵) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
1612con2i 567 . . . 4 (𝐵𝐴 → ¬ 𝐴 = 𝐵)
1716adantl 266 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ 𝐴 = 𝐵)
18 simpr 107 . . . . 5 ((¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴) → ¬ 𝐵𝐴)
1918con2i 567 . . . 4 (𝐵𝐴 → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2019adantl 266 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → ¬ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴))
2117, 202falsed 628 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
22 nntri3or 6102 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
239, 15, 21, 22mpjao3dan 1213 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341
This theorem is referenced by:  frec2uzf1od  9355
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