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Theorem atnlej1 38336
Description: If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
Hypotheses
Ref Expression
atnlej.l ≀ = (leβ€˜πΎ)
atnlej.j ∨ = (joinβ€˜πΎ)
atnlej.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atnlej1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑄)

Proof of Theorem atnlej1
StepHypRef Expression
1 hllat 38319 . . 3 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1133 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ Lat)
3 simp21 1206 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ 𝐴)
4 eqid 2732 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 atnlej.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38245 . . 3 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
73, 6syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
8 simp22 1207 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ 𝐴)
94, 5atbase 38245 . . 3 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
108, 9syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
11 simp23 1208 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ 𝐴)
124, 5atbase 38245 . . 3 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
1311, 12syl 17 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
14 simp3 1138 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅))
15 atnlej.l . . 3 ≀ = (leβ€˜πΎ)
16 atnlej.j . . 3 ∨ = (joinβ€˜πΎ)
174, 15, 16latnlej1l 18412 . 2 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑄)
182, 7, 10, 13, 14, 17syl131anc 1383 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  joincjn 18266  Latclat 18386  Atomscatm 38219  HLchlt 38306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-lub 18301  df-join 18303  df-lat 18387  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307
This theorem is referenced by:  4atlem0be  38552  dalem5  38624  dalem-cly  38628  4atexlemex6  39031  cdleme00a  39166  cdleme21a  39282  cdleme21b  39283  cdleme21c  39284
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