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Theorem atnle0 35118
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 35110 . . 3 (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
21adantr 466 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝐾 ∈ Poset)
3 eqid 2771 . . . 4 (Base‘𝐾) = (Base‘𝐾)
4 atnle0.z . . . 4 0 = (0.‘𝐾)
53, 4atl0cl 35112 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
65adantr 466 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ∈ (Base‘𝐾))
7 atnle0.a . . . 4 𝐴 = (Atoms‘𝐾)
83, 7atbase 35098 . . 3 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
98adantl 467 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ∈ (Base‘𝐾))
10 eqid 2771 . . 3 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
114, 10, 7atcvr0 35097 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 0 ( ⋖ ‘𝐾)𝑃)
12 atnle0.l . . 3 = (le‘𝐾)
133, 12, 10cvrnle 35089 . 2 (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 0 )
142, 6, 9, 11, 13syl31anc 1479 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  Basecbs 16064  lecple 16156  Posetcpo 17148  0.cp0 17245  ccvr 35071  Atomscatm 35072  AtLatcal 35073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-preset 17136  df-poset 17154  df-plt 17166  df-glb 17183  df-p0 17247  df-lat 17254  df-covers 35075  df-ats 35076  df-atl 35107
This theorem is referenced by:  pmap0  35573  trlnle  35995  cdlemg27b  36505
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