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Theorem trleile 32719
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 2728 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2tleile 18420 . . 3 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(leβ€˜πΎ)π‘Œ ∨ π‘Œ(leβ€˜πΎ)𝑋))
4 3simpc 1147 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
5 brxp 5731 . . . . . 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 5204 . . . . . 6 (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ 𝑋(𝐡 Γ— 𝐡)π‘Œ))
87rbaib 537 . . . . 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 460 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
11 brxp 5731 . . . . . 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 5204 . . . . . 6 (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ (π‘Œ(leβ€˜πΎ)𝑋 ∧ π‘Œ(𝐡 Γ— 𝐡)𝑋))
1413rbaib 537 . . . . 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 916 . . 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 5158 . . 3 (𝑋 ≀ π‘Œ ↔ 𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ)
2018breqi 5158 . . 3 (π‘Œ ≀ 𝑋 ↔ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋)
2119, 20orbi12i 912 . 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 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3948   class class class wbr 5152   Γ— cxp 5680  β€˜cfv 6553  Basecbs 17187  lecple 17247  Tosetctos 18415
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-iota 6505  df-fv 6561  df-toset 18416
This theorem is referenced by:  ordtrest2NEWlem  33556  ordtconnlem1  33558
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