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Theorem latasym 18488
Description: A lattice ordering is asymmetric. (eqss 3999 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
latref.b 𝐵 = (Base‘𝐾)
latref.l = (le‘𝐾)
Assertion
Ref Expression
latasym ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))

Proof of Theorem latasym
StepHypRef Expression
1 latref.b . . 3 𝐵 = (Base‘𝐾)
2 latref.l . . 3 = (le‘𝐾)
31, 2latasymb 18487 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
43biimpd 229 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  Latclat 18476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-iota 6514  df-fv 6569  df-proset 18340  df-poset 18359  df-lat 18477
This theorem is referenced by: (None)
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