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Theorem latasym 18405
Description: A lattice ordering is asymmetric. (eqss 3992 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
latref.b 𝐡 = (Baseβ€˜πΎ)
latref.l ≀ = (leβ€˜πΎ)
Assertion
Ref Expression
latasym ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) β†’ 𝑋 = π‘Œ))

Proof of Theorem latasym
StepHypRef Expression
1 latref.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 latref.l . . 3 ≀ = (leβ€˜πΎ)
31, 2latasymb 18404 . 2 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ 𝑋 = π‘Œ))
43biimpd 228 1 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) β†’ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6536  Basecbs 17150  lecple 17210  Latclat 18393
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6488  df-fv 6544  df-proset 18257  df-poset 18275  df-lat 18394
This theorem is referenced by: (None)
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