Theorem islln 38365

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	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 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.

Step	Hyp	Ref	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)
5	1, 2, 3, 4	llnset 38364	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
6	5	eleq2d 2819	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}))
7		breq2 5151	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑝𝐶𝑥 ↔ 𝑝𝐶𝑋))
8	7	rexbidv 3178	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑝 ∈ 𝐴 𝑝𝐶𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
9	8	elrab 3682	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
10	6, 9	bitrdi 286	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432   class class class wbr 5147  ‘cfv 6540  Basecbs 17140   ⋖ ccvr 38120  Atomscatm 38121  LLinesclln 38350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-llines 38357

This theorem is referenced by:  islln4  38366  llni  38367  llnbase  38368  llnnleat  38372