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Theorem trleile 30968
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
Hypotheses
Ref Expression
trleile.b 𝐵 = (Base‘𝐾)
trleile.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
Assertion
Ref Expression
trleile ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Proof of Theorem trleile
StepHypRef Expression
1 trleile.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2737 . . . 4 (le‘𝐾) = (le‘𝐾)
31, 2tleile 17927 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋))
4 3simpc 1152 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5 brxp 5598 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
64, 5sylibr 237 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 5105 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 542 . . . . 5 (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑋(le‘𝐾)𝑌))
96, 8syl 17 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑋(le‘𝐾)𝑌))
104ancomd 465 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐵𝑋𝐵))
11 brxp 5598 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
1210, 11sylibr 237 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13 brin 5105 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1413rbaib 542 . . . . 5 (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋𝑌(le‘𝐾)𝑋))
1512, 14syl 17 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋𝑌(le‘𝐾)𝑋))
169, 15orbi12d 919 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋) ↔ (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋)))
173, 16mpbird 260 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
18 trleile.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
1918breqi 5059 . . 3 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
2018breqi 5059 . . 3 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
2119, 20orbi12i 915 . 2 ((𝑋 𝑌𝑌 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
2217, 21sylibr 237 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847  w3a 1089   = wceq 1543  wcel 2110  cin 3865   class class class wbr 5053   × cxp 5549  cfv 6380  Basecbs 16760  lecple 16809  Tosetctos 17922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-iota 6338  df-fv 6388  df-toset 17923
This theorem is referenced by:  ordtrest2NEWlem  31586  ordtconnlem1  31588
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