Theorem perpin 28960

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 4302	. . 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 2769	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
5		eqid 2769	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
6		eqid 2769	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
7		eqid 2769	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
8		perpin.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8, 3	perpln1 28945	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (LineG‘𝐺))
10	7, 8, 3	perpln2 28946	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ran (LineG‘𝐺))
11	4, 5, 6, 7, 8, 9, 10	isperp 28947	. . 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 3179	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∩ cin 3912  ∅c0 4294   class class class wbr 5110  ‘cfv 6533  ⟨“cs3 14875  Basecbs 17265  distcds 17315  TarskiGcstrkg 28658  Itvcitv 28664  LineGclng 28665  ∟Gcrag 28928  ⟂Gcperpg 28930

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-sbc 3754  df-csb 3862  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-perpg 28931

This theorem is referenced by:  perpprlng  29149