Theorem islln 39966

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 39965	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
6	5	eleq2d 2823	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}))
7		breq2 5090	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑝𝐶𝑥 ↔ 𝑝𝐶𝑋))
8	7	rexbidv 3162	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑝 ∈ 𝐴 𝑝𝐶𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
9	8	elrab 3635	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
10	6, 9	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390   class class class wbr 5086  ‘cfv 6492  Basecbs 17170   ⋖ ccvr 39722  Atomscatm 39723  LLinesclln 39951

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-llines 39958

This theorem is referenced by:  islln4  39967  llni  39968  llnbase  39969  llnnleat  39973