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Theorem trleile 30006
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 2771 . . . 4 (le‘𝐾) = (le‘𝐾)
31, 2tleile 30001 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋))
4 3simpc 1146 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5 brxp 5287 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
64, 5sylibr 224 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 4838 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 528 . . . . 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 453 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐵𝑋𝐵))
11 brxp 5287 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
1210, 11sylibr 224 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
13 brin 4838 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1413rbaib 528 . . . . 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 902 . . 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 4792 . . 3 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
2018breqi 4792 . . 3 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
2119, 20orbi12i 898 . 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  wa 382  wo 834  w3a 1071   = wceq 1631  wcel 2145  cin 3722   class class class wbr 4786   × cxp 5247  cfv 6031  Basecbs 16064  lecple 16156  Tosetctos 17241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-iota 5994  df-fv 6039  df-toset 17242
This theorem is referenced by:  ordtrest2NEWlem  30308  ordtconnlem1  30310
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