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Theorem islln4 38980
Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐡 = (Baseβ€˜πΎ)
llnset.c 𝐢 = ( β‹– β€˜πΎ)
llnset.a 𝐴 = (Atomsβ€˜πΎ)
llnset.n 𝑁 = (LLinesβ€˜πΎ)
Assertion
Ref Expression
islln4 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
Distinct variable groups:   𝐴,𝑝   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐡(𝑝)   𝐢(𝑝)   𝐷(𝑝)   𝑁(𝑝)

Proof of Theorem islln4
StepHypRef Expression
1 llnset.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 llnset.c . . 3 𝐢 = ( β‹– β€˜πΎ)
3 llnset.a . . 3 𝐴 = (Atomsβ€˜πΎ)
4 llnset.n . . 3 𝑁 = (LLinesβ€˜πΎ)
51, 2, 3, 4islln 38979 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)))
65baibd 539 1 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067   class class class wbr 5148  β€˜cfv 6548  Basecbs 17180   β‹– ccvr 38734  Atomscatm 38735  LLinesclln 38964
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-sep 5299  ax-nul 5306  ax-pr 5429
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-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  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-iota 6500  df-fun 6550  df-fv 6556  df-llines 38971
This theorem is referenced by:  islln3  38983  llncmp  38995
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