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Theorem trleile 32643
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 2726 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
31, 2tleile 18383 . . 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 5718 . . . . . 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 5193 . . . . . 6 (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ 𝑋(𝐡 Γ— 𝐡)π‘Œ))
87rbaib 538 . . . . 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 461 . . . . . 6 ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
11 brxp 5718 . . . . . 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 5193 . . . . . 6 (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ (π‘Œ(leβ€˜πΎ)𝑋 ∧ π‘Œ(𝐡 Γ— 𝐡)𝑋))
1413rbaib 538 . . . . 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 915 . . 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 257 . 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 5147 . . 3 (𝑋 ≀ π‘Œ ↔ 𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ)
2018breqi 5147 . . 3 (π‘Œ ≀ 𝑋 ↔ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋)
2119, 20orbi12i 911 . 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   class class class wbr 5141   Γ— cxp 5667  β€˜cfv 6536  Basecbs 17150  lecple 17210  Tosetctos 18378
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-iota 6488  df-fv 6544  df-toset 18379
This theorem is referenced by:  ordtrest2NEWlem  33431  ordtconnlem1  33433
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