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Theorem latasymb 18437
Description: A lattice ordering is asymmetric. (eqss 3992 analog.) (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
latref.b 𝐵 = (Base‘𝐾)
latref.l = (le‘𝐾)
Assertion
Ref Expression
latasymb ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Proof of Theorem latasymb
StepHypRef Expression
1 latpos 18433 . 2 (𝐾 ∈ Lat → 𝐾 ∈ Poset)
2 latref.b . . 3 𝐵 = (Base‘𝐾)
3 latref.l . . 3 = (le‘𝐾)
42, 3posasymb 18314 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
51, 4syl3an1 1160 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  Posetcpo 18302  Latclat 18426
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 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-dm 5688  df-iota 6501  df-fv 6557  df-proset 18290  df-poset 18308  df-lat 18427
This theorem is referenced by:  latasym  18438  latasymd  18440  lubun  18510  cmtbr4N  38857  cvlexchb1  38932  hlateq  39002  cvratlem  39024  cvrat3  39045  pmap11  39365  cdleme50eq  40144  dia11N  40651  dib11N  40763  dih11  40868
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