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Theorem posrasymb 32130
Description: A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
posrasymb.b 𝐡 = (Baseβ€˜πΎ)
posrasymb.l ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
Assertion
Ref Expression
posrasymb ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ 𝑋 = π‘Œ))
Proof of Theorem posrasymb
StepHypRef Expression
1 posrasymb.l . . . . 5 ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))
21breqi 5154 . . . 4 (𝑋 ≀ π‘Œ ↔ 𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ)
3 simp2 1137 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
4 simp3 1138 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
5 brxp 5725 . . . . . 6 (𝑋(𝐡 Γ— 𝐡)π‘Œ ↔ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
63, 4, 5sylanbrc 583 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋(𝐡 Γ— 𝐡)π‘Œ)
7 brin 5200 . . . . . 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 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))π‘Œ ↔ 𝑋(leβ€˜πΎ)π‘Œ))
102, 9bitrid 282 . . 3 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋(leβ€˜πΎ)π‘Œ))
111breqi 5154 . . . 4 (π‘Œ ≀ 𝑋 ↔ π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋)
12 brxp 5725 . . . . . 6 (π‘Œ(𝐡 Γ— 𝐡)𝑋 ↔ (π‘Œ ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
134, 3, 12sylanbrc 583 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ(𝐡 Γ— 𝐡)𝑋)
14 brin 5200 . . . . . 6 (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ (π‘Œ(leβ€˜πΎ)𝑋 ∧ π‘Œ(𝐡 Γ— 𝐡)𝑋))
1514rbaib 539 . . . . 5 (π‘Œ(𝐡 Γ— 𝐡)𝑋 β†’ (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ π‘Œ(leβ€˜πΎ)𝑋))
1613, 15syl 17 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))𝑋 ↔ π‘Œ(leβ€˜πΎ)𝑋))
1711, 16bitrid 282 . . 3 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘Œ ≀ 𝑋 ↔ π‘Œ(leβ€˜πΎ)𝑋))
1810, 17anbi12d 631 . 2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ (𝑋(leβ€˜πΎ)π‘Œ ∧ π‘Œ(leβ€˜πΎ)𝑋)))
19 posrasymb.b . . 3 𝐡 = (Baseβ€˜πΎ)
20 eqid 2732 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2119, 20posasymb 18271 . 2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋(leβ€˜πΎ)π‘Œ ∧ π‘Œ(leβ€˜πΎ)𝑋) ↔ 𝑋 = π‘Œ))
2218, 21bitrd 278 1 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3947   class class class wbr 5148   Γ— cxp 5674  β€˜cfv 6543  Basecbs 17143  lecple 17203  Posetcpo 18259
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 5299  ax-nul 5306  ax-pr 5427
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-iota 6495  df-fv 6551  df-proset 18247  df-poset 18265
This theorem is referenced by:  ordtconnlem1  32899
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