Theorem prlngin0 29428

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 2760	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		prlngin0.p	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
5		prlngin0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29422	. . . . 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 2960	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  ran crn 5651  ‘cfv 6539  LineGclng 28902  hlGcplng 29257  parlnGcprlng 29420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-prlng 29421

This theorem is used by:  prlngpln3  29433  prlngmolem1  29436  prlngmolem2  29437  prlngmo2  29440  prlngpln4  29442  prlnginn0  29444  prlngsymquadlem  29447