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Theorem posrasymb 32106
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 5150 . . . 4 (𝑋 𝑌𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌)
3 simp2 1138 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
4 simp3 1139 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
5 brxp 5720 . . . . . 6 (𝑋(𝐵 × 𝐵)𝑌 ↔ (𝑋𝐵𝑌𝐵))
63, 4, 5sylanbrc 584 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑋(𝐵 × 𝐵)𝑌)
7 brin 5196 . . . . . 6 (𝑋((le‘𝐾) ∩ (𝐵 × 𝐵))𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋(𝐵 × 𝐵)𝑌))
87rbaib 540 . . . . 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, 9bitrid 283 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑋(le‘𝐾)𝑌))
111breqi 5150 . . . 4 (𝑌 𝑋𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋)
12 brxp 5720 . . . . . 6 (𝑌(𝐵 × 𝐵)𝑋 ↔ (𝑌𝐵𝑋𝐵))
134, 3, 12sylanbrc 584 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → 𝑌(𝐵 × 𝐵)𝑋)
14 brin 5196 . . . . . 6 (𝑌((le‘𝐾) ∩ (𝐵 × 𝐵))𝑋 ↔ (𝑌(le‘𝐾)𝑋𝑌(𝐵 × 𝐵)𝑋))
1514rbaib 540 . . . . 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, 16bitrid 283 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋𝑌(le‘𝐾)𝑋))
1810, 17anbi12d 632 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ (𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋)))
19 posrasymb.b . . 3 𝐵 = (Base‘𝐾)
20 eqid 2733 . . 3 (le‘𝐾) = (le‘𝐾)
2119, 20posasymb 18259 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌))
2218, 21bitrd 279 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cin 3945   class class class wbr 5144   × cxp 5670  cfv 6535  Basecbs 17131  lecple 17191  Posetcpo 18247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-xp 5678  df-iota 6487  df-fv 6543  df-proset 18235  df-poset 18253
This theorem is referenced by:  ordtconnlem1  32835
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