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Theorem atnle0 39291
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 39283 . . 3 (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
21adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝐾 ∈ Poset)
3 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
4 atnle0.z . . . 4 0 = (0.‘𝐾)
53, 4atl0cl 39285 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
65adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ∈ (Base‘𝐾))
7 atnle0.a . . . 4 𝐴 = (Atoms‘𝐾)
83, 7atbase 39271 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
98adantl 481 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ∈ (Base‘𝐾))
10 eqid 2735 . . 3 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
114, 10, 7atcvr0 39270 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ( ⋖ ‘𝐾)𝑃)
12 atnle0.l . . 3 = (le‘𝐾)
133, 12, 10cvrnle 39262 . 2 (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 0 )
142, 6, 9, 11, 13syl31anc 1372 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  0.cp0 18481  ccvr 39244  Atomscatm 39245  AtLatcal 39246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-proset 18352  df-poset 18371  df-plt 18388  df-glb 18405  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280
This theorem is referenced by:  pmap0  39748  trlnle  40169  cdlemg27b  40679
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