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Theorem atnle0 38781
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 38773 . . 3 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Poset)
21adantr 480 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝐾 ∈ Poset)
3 eqid 2728 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 atnle0.z . . . 4 0 = (0.β€˜πΎ)
53, 4atl0cl 38775 . . 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 38761 . . 3 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
98adantl 481 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
10 eqid 2728 . . 3 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
114, 10, 7atcvr0 38760 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 0 ( β‹– β€˜πΎ)𝑃)
12 atnle0.l . . 3 ≀ = (leβ€˜πΎ)
133, 12, 10cvrnle 38752 . 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 1371 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 1534   ∈ wcel 2099   class class class wbr 5148  β€˜cfv 6548  Basecbs 17180  lecple 17240  Posetcpo 18299  0.cp0 18415   β‹– ccvr 38734  Atomscatm 38735  AtLatcal 38736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-proset 18287  df-poset 18305  df-plt 18322  df-glb 18339  df-p0 18417  df-lat 18424  df-covers 38738  df-ats 38739  df-atl 38770
This theorem is referenced by:  pmap0  39238  trlnle  39659  cdlemg27b  40169
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