Theorem isline 40571

Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)

Hypotheses

Ref	Expression
isline.l	⊢ ≤ = (le‘𝐾)
isline.j	⊢ ∨ = (join‘𝐾)
isline.a	⊢ 𝐴 = (Atoms‘𝐾)
isline.n	⊢ 𝑁 = (Lines‘𝐾)

Assertion

Ref	Expression
isline	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))

Distinct variable groups:   𝑞,𝑝,𝑟,𝐴   𝐾,𝑝,𝑞,𝑟   𝑋,𝑞,𝑟

Allowed substitution hints:   𝐷(𝑟, 𝑞, 𝑝)   ∨ (𝑟, 𝑞, 𝑝)   ≤ (𝑟, 𝑞, 𝑝)   𝑁(𝑟, 𝑞, 𝑝)   𝑋(𝑝)

Proof of Theorem isline

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isline.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		isline.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		isline.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4		isline.n	. . . 4 ⊢ 𝑁 = (Lines‘𝐾)
5	1, 2, 3, 4	lineset 40570	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
6	5	eleq2d 2851	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})}))
7	3	fvexi 6899	. . . . . . . 8 ⊢ 𝐴 ∈ V
8	7	rabex 5311	. . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V
9		eleq1 2853	. . . . . . 7 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V))
10	8, 9	mpbiri 261	. . . . . 6 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → 𝑋 ∈ V)
11	10	adantl 487	. . . . 5 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V)
12	11	a1i 11	. . . 4 ⊢ ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V))
13	12	rexlimivv 3209	. . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V)
14		eqeq1 2769	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
15	14	anbi2d 642	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
16	15	2rexbidv 3232	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
17	13, 16	elab3 3647	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18	6, 17	bitrdi 290	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∃wrex 3091  {crab 3418  Vcvv 3457   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  lecple 17339  joincjn 18389  Atomscatm 40095  Linesclines 40326

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-lines 40333

This theorem is used by:  islinei  40572  linepsubN  40584  isline2  40606