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Theorem islln 37998
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 37997 . . 3 (𝐾 ∈ 𝐷 β†’ 𝑁 = {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯})
65eleq2d 2824 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {π‘₯ ∈ 𝐡 ∣ βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯}))
7 breq2 5114 . . . 4 (π‘₯ = 𝑋 β†’ (𝑝𝐢π‘₯ ↔ 𝑝𝐢𝑋))
87rexbidv 3176 . . 3 (π‘₯ = 𝑋 β†’ (βˆƒπ‘ ∈ 𝐴 𝑝𝐢π‘₯ ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
98elrab 3650 . 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410   class class class wbr 5110  β€˜cfv 6501  Basecbs 17090   β‹– ccvr 37753  Atomscatm 37754  LLinesclln 37983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-llines 37990
This theorem is referenced by:  islln4  37999  llni  38000  llnbase  38001  llnnleat  38005
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