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Theorem llnbase 34614
Description: A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnbase.b 𝐵 = (Base‘𝐾)
llnbase.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llnbase (𝑋𝑁𝑋𝐵)

Proof of Theorem llnbase
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 n0i 3912 . . . 4 (𝑋𝑁 → ¬ 𝑁 = ∅)
2 llnbase.n . . . . 5 𝑁 = (LLines‘𝐾)
32eqeq1i 2625 . . . 4 (𝑁 = ∅ ↔ (LLines‘𝐾) = ∅)
41, 3sylnib 318 . . 3 (𝑋𝑁 → ¬ (LLines‘𝐾) = ∅)
5 fvprc 6172 . . 3 𝐾 ∈ V → (LLines‘𝐾) = ∅)
64, 5nsyl2 142 . 2 (𝑋𝑁𝐾 ∈ V)
7 llnbase.b . . . 4 𝐵 = (Base‘𝐾)
8 eqid 2620 . . . 4 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
9 eqid 2620 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
107, 8, 9, 2islln 34611 . . 3 (𝐾 ∈ V → (𝑋𝑁 ↔ (𝑋𝐵 ∧ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)))
1110simprbda 652 . 2 ((𝐾 ∈ V ∧ 𝑋𝑁) → 𝑋𝐵)
126, 11mpancom 702 1 (𝑋𝑁𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wrex 2910  Vcvv 3195  c0 3907   class class class wbr 4644  cfv 5876  Basecbs 15838  ccvr 34368  Atomscatm 34369  LLinesclln 34596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-llines 34603
This theorem is referenced by:  islln2  34616  llnnleat  34618  llnneat  34619  atcvrlln2  34624  llnexatN  34626  llncmp  34627  2llnmat  34629  islpln3  34638  llnmlplnN  34644  lplnle  34645  lplnnle2at  34646  llncvrlpln2  34662  llncvrlpln  34663  2llnmj  34665  lplncmp  34667  lplnexatN  34668  lplnexllnN  34669  2llnm2N  34673  2llnm3N  34674  2llnm4  34675  2llnmeqat  34676  dalem21  34799  dalem54  34831  dalem55  34832  dalem57  34834  dalem60  34837  llnexchb2lem  34973  llnexchb2  34974  llnexch2N  34975
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