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Theorem nntri2 6513
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
Assertion
Ref Expression
nntri2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Proof of Theorem nntri2
StepHypRef Expression
1 elirr 4555 . . . . 5 ¬ 𝐴𝐴
2 eleq2 2253 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2mtbii 675 . . . 4 (𝐴 = 𝐵 → ¬ 𝐴𝐵)
43con2i 628 . . 3 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
5 en2lp 4568 . . . 4 ¬ (𝐴𝐵𝐵𝐴)
65imnani 692 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
7 ioran 753 . . 3 (¬ (𝐴 = 𝐵𝐵𝐴) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐵𝐴))
84, 6, 7sylanbrc 417 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐵𝐵𝐴))
9 nntri3or 6512 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
10 3orass 983 . . . . 5 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ (𝐴𝐵 ∨ (𝐴 = 𝐵𝐵𝐴)))
119, 10sylib 122 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ∨ (𝐴 = 𝐵𝐵𝐴)))
1211orcomd 730 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = 𝐵𝐵𝐴) ∨ 𝐴𝐵))
1312ord 725 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐴 = 𝐵𝐵𝐴) → 𝐴𝐵))
148, 13impbid2 143 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  w3o 979   = wceq 1364  wcel 2160  ωcom 4604
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 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
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 4381  df-on 4383  df-suc 4386  df-iom 4605
This theorem is referenced by:  nnaord  6528  nnmord  6536  pitric  7338
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