Theorem islln 36675

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 36674	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥})
6	5	eleq2d 2897	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥}))
7		breq2 5063	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑝𝐶𝑥 ↔ 𝑝𝐶𝑋))
8	7	rexbidv 3296	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑝 ∈ 𝐴 𝑝𝐶𝑥 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
9	8	elrab 3676	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
10	6, 9	syl6bb 289	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∃wrex 3138  {crab 3141   class class class wbr 5059  ‘cfv 6348  Basecbs 16478   ⋖ ccvr 36431  Atomscatm 36432  LLinesclln 36660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-llines 36667

This theorem is referenced by:  islln4  36676  llni  36677  llnbase  36678  llnnleat  36682