Theorem prlngin0 29223

Description: Two parallel lines do not intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
prlngin0.l	⊢ 𝐿 = (LineG‘𝐺)
prlngin0.p	⊢ ∥ = (parlnG‘𝐺)
prlngin0.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
prlngin0.1	⊢ (𝜑 → 𝐴 ∥ 𝐵)
prlngin0.2	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
prlngin0	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Proof of Theorem prlngin0

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngin0.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∥ 𝐵)
2		prlngin0.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
3		eqid 2765	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		prlngin0.p	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
5		prlngin0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29217	. . . . 5 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
7	1, 6	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
8	7	simprd 501	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
9		prlngin0.2	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
10	9	neneqd 2965	. . 3 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
11	8, 10	orcnd 892	. 2 ⊢ (𝜑 → (∃ℎ ∈ ran (hlG‘𝐺)(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))
12	11	simprd 501	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   class class class wbr 5111  ran crn 5664  ‘cfv 6540  LineGclng 28732  hlGcplng 29084  parlnGcprlng 29215

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-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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-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-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-prlng 29216

This theorem is used by:  prlngpln3  29228  prlngmolem1  29231  prlngmolem2  29232  prlngmo2  29235  prlngpln4  29237  prlnginn0  29239  prlngsymquadlem  29242