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Theorem atnle0 37817
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 37809 . . 3 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Poset)
21adantr 482 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝐾 ∈ Poset)
3 eqid 2733 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 atnle0.z . . . 4 0 = (0.β€˜πΎ)
53, 4atl0cl 37811 . . 3 (𝐾 ∈ AtLat β†’ 0 ∈ (Baseβ€˜πΎ))
65adantr 482 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 0 ∈ (Baseβ€˜πΎ))
7 atnle0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
83, 7atbase 37797 . . 3 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
98adantl 483 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
10 eqid 2733 . . 3 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
114, 10, 7atcvr0 37796 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 0 ( β‹– β€˜πΎ)𝑃)
12 atnle0.l . . 3 ≀ = (leβ€˜πΎ)
133, 12, 10cvrnle 37788 . 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 1374 1 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ Β¬ 𝑃 ≀ 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  Basecbs 17088  lecple 17145  Posetcpo 18201  0.cp0 18317   β‹– ccvr 37770  Atomscatm 37771  AtLatcal 37772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-proset 18189  df-poset 18207  df-plt 18224  df-glb 18241  df-p0 18319  df-lat 18326  df-covers 37774  df-ats 37775  df-atl 37806
This theorem is referenced by:  pmap0  38274  trlnle  38695  cdlemg27b  39205
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