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Theorem islln 36801
 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 36800 . . 3 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
65eleq2d 2878 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥}))
7 breq2 5037 . . . 4 (𝑥 = 𝑋 → (𝑝𝐶𝑥𝑝𝐶𝑋))
87rexbidv 3259 . . 3 (𝑥 = 𝑋 → (∃𝑝𝐴 𝑝𝐶𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
98elrab 3631 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋))
106, 9syl6bb 290 1 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∃wrex 3110  {crab 3113   class class class wbr 5033  ‘cfv 6328  Basecbs 16479   ⋖ ccvr 36557  Atomscatm 36558  LLinesclln 36786 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-llines 36793 This theorem is referenced by:  islln4  36802  llni  36803  llnbase  36804  llnnleat  36808
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