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Theorem posrasymb 28988
 Description: A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
posrasymb.b 𝐵 = (Base‘𝐾)
posrasymb.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
Assertion
Ref Expression
posrasymb ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Proof of Theorem posrasymb
StepHypRef Expression
1 posrasymb.l . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
21breqi 4589 . . . 4 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
3 simp2 1055 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
4 simp3 1056 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
5 brxp 5071 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
63, 4, 5sylanbrc 695 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 4634 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 945 . . . . 5 (𝑋(𝐵 × 𝐵)𝑌 → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑋(le‘𝐾)𝑌))
96, 8syl 17 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌𝑋(le‘𝐾)𝑌))
102, 9syl5bb 271 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑋(le‘𝐾)𝑌))
111breqi 4589 . . . 4 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
12 brxp 5071 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
134, 3, 12sylanbrc 695 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
14 brin 4634 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1514rbaib 945 . . . . 5 (𝑌(𝐵 × 𝐵)𝑋 → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋𝑌(le‘𝐾)𝑋))
1613, 15syl 17 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋𝑌(le‘𝐾)𝑋))
1711, 16syl5bb 271 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋𝑌(le‘𝐾)𝑋))
1810, 17anbi12d 743 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋)))
19 posrasymb.b . . 3 𝐵 = (Base‘𝐾)
20 eqid 2610 . . 3 (le‘𝐾) = (le‘𝐾)
2119, 20posasymb 16775 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌))
2218, 21bitrd 267 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   class class class wbr 4583   × cxp 5036  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812  df-preset 16751  df-poset 16769 This theorem is referenced by:  ordtconlem1  29298
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