Theorem perpin 28986

Description: If two lines 𝐴 and 𝐵 are perpendicular, then they intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
perpin.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
perpin.2	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Assertion

Ref	Expression
perpin	⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

Proof of Theorem perpin

Dummy variables 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ne0i 4295	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ≠ ∅)
2	1	ad2antlr 739	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → (𝐴 ∩ 𝐵) ≠ ∅)
3		perpin.2	. . 3 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)
4		eqid 2763	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
5		eqid 2763	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
6		eqid 2763	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
7		eqid 2763	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
8		perpin.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8, 3	perpln1 28971	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (LineG‘𝐺))
10	7, 8, 3	perpln2 28972	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ran (LineG‘𝐺))
11	4, 5, 6, 7, 8, 9, 10	isperp 28973	. . 3 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
12	3, 11	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
13	2, 12	r19.29a 3173	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ∩ cin 3905  ∅c0 4287   class class class wbr 5110  ‘cfv 6538  ⟨“cs3 14881  Basecbs 17270  distcds 17320  TarskiGcstrkg 28677  Itvcitv 28683  LineGclng 28684  ∟Gcrag 28954  ⟂Gcperpg 28956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-perpg 28957

This theorem is referenced by:  perpprlng  29181