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Theorem islln4 35527
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
islln4 ((𝐾𝐷𝑋𝐵) → (𝑋𝑁 ↔ ∃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 islln4
StepHypRef Expression
1 llnset.b . . 3 𝐵 = (Base‘𝐾)
2 llnset.c . . 3 𝐶 = ( ⋖ ‘𝐾)
3 llnset.a . . 3 𝐴 = (Atoms‘𝐾)
4 llnset.n . . 3 𝑁 = (LLines‘𝐾)
51, 2, 3, 4islln 35526 . 2 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
65baibd 536 1 ((𝐾𝐷𝑋𝐵) → (𝑋𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3091   class class class wbr 4844  cfv 6102  Basecbs 16183  ccvr 35282  Atomscatm 35283  LLinesclln 35511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-llines 35518
This theorem is referenced by:  islln3  35530  llncmp  35542
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