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Theorem trleile 29443
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 2626 . . . 4 (le‘𝐾) = (le‘𝐾)
31, 2tleile 29438 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋))
4 3simpc 1058 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5 brxp 5112 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
64, 5sylibr 224 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 4669 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 946 . . . . 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 467 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐵𝑋𝐵))
11 brxp 5112 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
1210, 11sylibr 224 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13 brin 4669 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1413rbaib 946 . . . . 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 745 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋) ↔ (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋)))
173, 16mpbird 247 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
18 trleile.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
1918breqi 4624 . . 3 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
2018breqi 4624 . . 3 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
2119, 20orbi12i 543 . 2 ((𝑋 𝑌𝑌 𝑋) ↔ (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋))
2217, 21sylibr 224 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  cin 3559   class class class wbr 4618   × cxp 5077  cfv 5850  Basecbs 15776  lecple 15864  Tosetctos 16949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-iota 5813  df-fv 5858  df-toset 16950
This theorem is referenced by:  ordtrest2NEWlem  29742  ordtconnlem1  29744
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