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Theorem trleile 32128
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
Hypotheses
Ref Expression
trleile.b 𝐡 = (Baseβ€˜πΎ)
trleile.l ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
Assertion
Ref Expression
trleile ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ∨ π‘Œ ≀ 𝑋))
Proof of Theorem trleile
StepHypRef Expression
1 trleile.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 eqid 2732 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2tleile 18370 . . 3 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(leβ€˜πΎ)π‘Œ ∨ π‘Œ(leβ€˜πΎ)𝑋))
4 3simpc 1150 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
5 brxp 5723 . . . . . 6 (𝑋(𝐡 Γ— 𝐡)π‘Œ ↔ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
64, 5sylibr 233 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋(𝐡 Γ— 𝐡)π‘Œ)
7 brin 5199 . . . . . 6 (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ 𝑋(𝐡 Γ— 𝐡)π‘Œ))
87rbaib 539 . . . . 5 (𝑋(𝐡 Γ— 𝐡)π‘Œ β†’ (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ 𝑋(leβ€˜πΎ)π‘Œ))
96, 8syl 17 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ 𝑋(leβ€˜πΎ)π‘Œ))
104ancomd 462 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
11 brxp 5723 . . . . . 6 (π‘Œ(𝐡 Γ— 𝐡)𝑋 ↔ (π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
1210, 11sylibr 233 . . . . 5 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ(𝐡 Γ— 𝐡)𝑋)
13 brin 5199 . . . . . 6 (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ (π‘Œ(leβ€˜πΎ)𝑋 ∧ π‘Œ(𝐡 Γ— 𝐡)𝑋))
1413rbaib 539 . . . . 5 (π‘Œ(𝐡 Γ— 𝐡)𝑋 β†’ (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ π‘Œ(leβ€˜πΎ)𝑋))
1512, 14syl 17 . . . 4 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ π‘Œ(leβ€˜πΎ)𝑋))
169, 15orbi12d 917 . . 3 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ∨ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋) ↔ (𝑋(leβ€˜πΎ)π‘Œ ∨ π‘Œ(leβ€˜πΎ)𝑋)))
173, 16mpbird 256 . 2 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ∨ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋))
18 trleile.l . . . 4 ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
1918breqi 5153 . . 3 (𝑋 ≀ π‘Œ ↔ 𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ)
2018breqi 5153 . . 3 (π‘Œ ≀ 𝑋 ↔ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋)
2119, 20orbi12i 913 . 2 ((𝑋 ≀ π‘Œ ∨ π‘Œ ≀ 𝑋) ↔ (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ∨ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋))
2217, 21sylibr 233 1 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ∨ π‘Œ ≀ 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946   class class class wbr 5147   Γ— cxp 5673  β€˜cfv 6540  Basecbs 17140  lecple 17200  Tosetctos 18365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-iota 6492  df-fv 6548  df-toset 18366
This theorem is referenced by:  ordtrest2NEWlem  32890  ordtconnlem1  32892
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