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Theorem atnlej1 38250
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 38233 . . 3 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1134 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ Lat)
3 simp21 1207 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ 𝐴)
4 eqid 2733 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 atnlej.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38159 . . 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 1208 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ 𝐴)
94, 5atbase 38159 . . 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 1209 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ 𝐴)
124, 5atbase 38159 . . 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 1139 . 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 18410 . 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 1384 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 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  Latclat 18384  Atomscatm 38133  HLchlt 38220
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-lub 18299  df-join 18301  df-lat 18385  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221
This theorem is referenced by:  4atlem0be  38466  dalem5  38538  dalem-cly  38542  4atexlemex6  38945  cdleme00a  39080  cdleme21a  39196  cdleme21b  39197  cdleme21c  39198
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