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Theorem atnle0 33614
 Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
atnle0.l = (le‘𝐾)
atnle0.z 0 = (0.‘𝐾)
atnle0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atnle0 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )

Proof of Theorem atnle0
StepHypRef Expression
1 atlpos 33606 . . 3 (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
21adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝐾 ∈ Poset)
3 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
4 atnle0.z . . . 4 0 = (0.‘𝐾)
53, 4atl0cl 33608 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
65adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ∈ (Base‘𝐾))
7 atnle0.a . . . 4 𝐴 = (Atoms‘𝐾)
83, 7atbase 33594 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
98adantl 481 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ∈ (Base‘𝐾))
10 eqid 2610 . . 3 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
114, 10, 7atcvr0 33593 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ( ⋖ ‘𝐾)𝑃)
12 atnle0.l . . 3 = (le‘𝐾)
133, 12, 10cvrnle 33585 . 2 (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 0 )
142, 6, 9, 11, 13syl31anc 1321 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  0.cp0 16860   ⋖ ccvr 33567  Atomscatm 33568  AtLatcal 33569 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-preset 16751  df-poset 16769  df-plt 16781  df-glb 16798  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603 This theorem is referenced by:  pmap0  34069  trlnle  34491  cdlemg27b  35002
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