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Theorem islln 38973
Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b 𝐡 = (Baseβ€˜πΎ)
llnset.c 𝐢 = ( β‹– β€˜πΎ)
llnset.a 𝐴 = (Atomsβ€˜πΎ)
llnset.n 𝑁 = (LLinesβ€˜πΎ)
Assertion
Ref Expression
islln (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)))
Distinct variable groups:   𝐴,𝑝   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐡(𝑝)   𝐢(𝑝)   𝐷(𝑝)   𝑁(𝑝)

Proof of Theorem islln
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 llnset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 llnset.c . . . 4 𝐢 = ( β‹– β€˜πΎ)
3 llnset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 llnset.n . . . 4 𝑁 = (LLinesβ€˜πΎ)
51, 2, 3, 4llnset 38972 . . 3 (𝐾 ∈ 𝐷 β†’ 𝑁 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
65eleq2d 2815 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯}))
7 breq2 5146 . . . 4 (π‘₯ = 𝑋 β†’ (𝑝𝐢π‘₯ ↔ 𝑝𝐢𝑋))
87rexbidv 3174 . . 3 (π‘₯ = 𝑋 β†’ (βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯ ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
98elrab 3681 . 2 (𝑋 ∈ {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯} ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
106, 9bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3066  {crab 3428   class class class wbr 5142  β€˜cfv 6542  Basecbs 17173   β‹– ccvr 38728  Atomscatm 38729  LLinesclln 38958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-llines 38965
This theorem is referenced by:  islln4  38974  llni  38975  llnbase  38976  llnnleat  38980
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