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Theorem cdleme0gN 39748
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l ≀ = (leβ€˜πΎ)
cdleme0.j ∨ = (joinβ€˜πΎ)
cdleme0.m ∧ = (meetβ€˜πΎ)
cdleme0.a 𝐴 = (Atomsβ€˜πΎ)
cdleme0.h 𝐻 = (LHypβ€˜πΎ)
cdleme0.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme0c.3 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
Assertion
Ref Expression
cdleme0gN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝑉 β‰  𝑄)

Proof of Theorem cdleme0gN
StepHypRef Expression
1 cdleme0.l . 2 ≀ = (leβ€˜πΎ)
2 cdleme0.j . 2 ∨ = (joinβ€˜πΎ)
3 cdleme0.m . 2 ∧ = (meetβ€˜πΎ)
4 cdleme0.a . 2 𝐴 = (Atomsβ€˜πΎ)
5 cdleme0.h . 2 𝐻 = (LHypβ€˜πΎ)
6 cdleme0c.3 . 2 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
71, 2, 3, 4, 5, 6cdleme0c 39742 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝑉 β‰  𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  lecple 17239  joincjn 18302  meetcmee 18303  Atomscatm 38791  HLchlt 38878  LHypclh 39513
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-lat 18423  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-lhyp 39517
This theorem is referenced by: (None)
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