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Theorem trleile 32787
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 2725 . . . 4 (le‘𝐾) = (le‘𝐾)
31, 2tleile 18416 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋))
4 3simpc 1147 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5 brxp 5727 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
64, 5sylibr 233 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 5201 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 537 . . . . 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 460 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐵𝑋𝐵))
11 brxp 5727 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
1210, 11sylibr 233 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13 brin 5201 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1413rbaib 537 . . . . 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 916 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋) ↔ (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋)))
173, 16mpbird 256 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
18 trleile.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
1918breqi 5155 . . 3 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
2018breqi 5155 . . 3 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
2119, 20orbi12i 912 . 2 ((𝑋 𝑌𝑌 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
2217, 21sylibr 233 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  cin 3943   class class class wbr 5149   × cxp 5676  cfv 6549  Basecbs 17183  lecple 17243  Tosetctos 18411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-iota 6501  df-fv 6557  df-toset 18412
This theorem is referenced by:  ordtrest2NEWlem  33654  ordtconnlem1  33656
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