Theorem prlngpln 29146

Description: Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

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

Assertion

Ref	Expression
prlngpln	⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))

Distinct variable groups:   𝐴,ℎ   𝐵,ℎ   ℎ,𝐸   ℎ,𝐺

Allowed substitution hints:   𝜑(ℎ)   ∥ (ℎ)   𝐿(ℎ)   𝑉(ℎ)

Proof of Theorem prlngpln

Step	Hyp	Ref	Expression
1		prlngpln.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∥ 𝐵)
2		prlngpln.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
3		prlngpln.e	. . . . . 6 ⊢ 𝐸 = (hlG‘𝐺)
4		prlngpln.p	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
5		prlngpln.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	2, 3, 4, 5	brprlng 29139	. . . . 5 ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
7	1, 6	mpbid 235	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))))
8	7	simprd 500	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))
9		prlngpln.2	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
10	9	neneqd 2969	. . 3 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
11	8, 10	orcnd 891	. 2 ⊢ (𝜑 → (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅))
12	11	simpld 499	1 ⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294   class class class wbr 5110  ran crn 5660  ‘cfv 6533  LineGclng 28665  hlGcplng 29009  parlnGcprlng 29137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fv 6541  df-prlng 29138

This theorem is referenced by: (None)