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Theorem islln 38840
Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐵 = (Base‘𝐾)
llnset.c 𝐶 = ( ⋖ ‘𝐾)
llnset.a 𝐴 = (Atoms‘𝐾)
llnset.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
islln (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
Distinct variable groups:   𝐴,𝑝   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐵(𝑝)   𝐶(𝑝)   𝐷(𝑝)   𝑁(𝑝)

Proof of Theorem islln
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 llnset.b . . . 4 𝐵 = (Base‘𝐾)
2 llnset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
3 llnset.a . . . 4 𝐴 = (Atoms‘𝐾)
4 llnset.n . . . 4 𝑁 = (LLines‘𝐾)
51, 2, 3, 4llnset 38839 . . 3 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
65eleq2d 2818 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥}))
7 breq2 5152 . . . 4 (𝑥 = 𝑋 → (𝑝𝐶𝑥𝑝𝐶𝑋))
87rexbidv 3177 . . 3 (𝑥 = 𝑋 → (∃𝑝𝐴 𝑝𝐶𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
98elrab 3683 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋))
106, 9bitrdi 287 1 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wrex 3069  {crab 3431   class class class wbr 5148  cfv 6543  Basecbs 17151  ccvr 38595  Atomscatm 38596  LLinesclln 38825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  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-iota 6495  df-fun 6545  df-fv 6551  df-llines 38832
This theorem is referenced by:  islln4  38841  llni  38842  llnbase  38843  llnnleat  38847
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