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Theorem cdlemesner 38047
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
cdlemesner.l = (le‘𝐾)
cdlemesner.j = (join‘𝐾)
cdlemesner.a 𝐴 = (Atoms‘𝐾)
cdlemesner.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
cdlemesner ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝑅)

Proof of Theorem cdlemesner
StepHypRef Expression
1 nbrne2 5073 . . 3 ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑅𝑆)
213ad2ant3 1137 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝑆)
32necomd 2996 1 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940   class class class wbr 5053  cfv 6380  (class class class)co 7213  lecple 16809  joincjn 17818  Atomscatm 37014  HLchlt 37101  LHypclh 37735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054
This theorem is referenced by:  cdlemeda  38049
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