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Theorem islln 33606
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 33605 . . 3 (𝐾𝐷𝑁 = {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥})
65eleq2d 2672 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥}))
7 breq2 4581 . . . 4 (𝑥 = 𝑋 → (𝑝𝐶𝑥𝑝𝐶𝑋))
87rexbidv 3033 . . 3 (𝑥 = 𝑋 → (∃𝑝𝐴 𝑝𝐶𝑥 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
98elrab 3330 . 2 (𝑋 ∈ {𝑥𝐵 ∣ ∃𝑝𝐴 𝑝𝐶𝑥} ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋))
106, 9syl6bb 274 1 (𝐾𝐷 → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝𝐴 𝑝𝐶𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  {crab 2899   class class class wbr 4577  cfv 5790  Basecbs 15641  ccvr 33363  Atomscatm 33364  LLinesclln 33591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-llines 33598
This theorem is referenced by:  islln4  33607  llni  33608  llnbase  33609  llnnleat  33613
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