Theorem isline 39911

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 39910	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝑁 = {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
6	5	eleq2d 2819	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ 𝑋 ∈ {𝑥 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})}))
7	3	fvexi 6845	. . . . . . . 8 ⊢ 𝐴 ∈ V
8	7	rabex 5281	. . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V
9		eleq1 2821	. . . . . . 7 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → (𝑋 ∈ V ↔ {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V))
10	8, 9	mpbiri 258	. . . . . 6 ⊢ (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} → 𝑋 ∈ V)
11	10	adantl 481	. . . . 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 3175	. . 3 ⊢ (∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑋 ∈ V)
14		eqeq1 2737	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
15	14	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑞 ≠ 𝑟 ∧ 𝑥 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
16	15	2rexbidv 3198	. . 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 3638	. 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 287	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711   ≠ wne 2929  ∃wrex 3057  {crab 3396  Vcvv 3437   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355  lecple 17175  joincjn 18225  Atomscatm 39435  Linesclines 39666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-lines 39673

This theorem is referenced by:  islinei  39912  linepsubN  39924  isline2  39946