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Theorem atnle0 38683
Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
atnle0.l ≀ = (leβ€˜πΎ)
atnle0.z 0 = (0.β€˜πΎ)
atnle0.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atnle0 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ Β¬ 𝑃 ≀ 0 )

Proof of Theorem atnle0
StepHypRef Expression
1 atlpos 38675 . . 3 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Poset)
21adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝐾 ∈ Poset)
3 eqid 2724 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 atnle0.z . . . 4 0 = (0.β€˜πΎ)
53, 4atl0cl 38677 . . 3 (𝐾 ∈ AtLat β†’ 0 ∈ (Baseβ€˜πΎ))
65adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 0 ∈ (Baseβ€˜πΎ))
7 atnle0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
83, 7atbase 38663 . . 3 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
98adantl 481 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
10 eqid 2724 . . 3 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
114, 10, 7atcvr0 38662 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 0 ( β‹– β€˜πΎ)𝑃)
12 atnle0.l . . 3 ≀ = (leβ€˜πΎ)
133, 12, 10cvrnle 38654 . 2 (((𝐾 ∈ Poset ∧ 0 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ)) ∧ 0 ( β‹– β€˜πΎ)𝑃) β†’ Β¬ 𝑃 ≀ 0 )
142, 6, 9, 11, 13syl31anc 1370 1 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ Β¬ 𝑃 ≀ 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5139  β€˜cfv 6534  Basecbs 17149  lecple 17209  Posetcpo 18268  0.cp0 18384   β‹– ccvr 38636  Atomscatm 38637  AtLatcal 38638
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-proset 18256  df-poset 18274  df-plt 18291  df-glb 18308  df-p0 18386  df-lat 18393  df-covers 38640  df-ats 38641  df-atl 38672
This theorem is referenced by:  pmap0  39140  trlnle  39561  cdlemg27b  40071
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