Theorem prlngin0 29479

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 2763	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		prlngin0.p	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
5		prlngin0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29473	. . . . 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 2964	. . 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 2147   ≠ wne 2959  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   class class class wbr 5108  ran crn 5659  ‘cfv 6566  LineGclng 28953  hlGcplng 29308  parlnGcprlng 29471

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-prlng 29472

This theorem is used by:  prlngpln3  29484  prlngmolem1  29487  prlngmolem2  29488  prlngmo2  29491  prlngpln4  29493  prlnginn0  29495  prlngsymquadlem  29498