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Theorem islln 35314
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 35313 . . 3 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
65eleq2d 2836 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥}))
7 breq2 4790 . . . 4 (𝑥 = 𝑋 → (𝑝𝐶𝑥𝑝𝐶𝑋))
87rexbidv 3200 . . 3 (𝑥 = 𝑋 → (∃𝑝𝐴 𝑝𝐶𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
98elrab 3515 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋))
106, 9syl6bb 276 1 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  {crab 3065   class class class wbr 4786  cfv 6031  Basecbs 16064  ccvr 35071  Atomscatm 35072  LLinesclln 35299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-llines 35306
This theorem is referenced by:  islln4  35315  llni  35316  llnbase  35317  llnnleat  35321
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