Theorem perpin 29034

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 4294	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ≠ ∅)
2	1	ad2antlr 740	. 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 2765	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
5		eqid 2765	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
6		eqid 2765	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
7		eqid 2765	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
8		perpin.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8, 3	perpln1 29019	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (LineG‘𝐺))
10	7, 8, 3	perpln2 29020	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ran (LineG‘𝐺))
11	4, 5, 6, 7, 8, 9, 10	isperp 29021	. . 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 3175	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∩ cin 3905  ∅c0 4286   class class class wbr 5111  ‘cfv 6540  ⟨“cs3 14898  Basecbs 17286  distcds 17336  TarskiGcstrkg 28725  Itvcitv 28731  LineGclng 28732  ∟Gcrag 29002  ⟂Gcperpg 29004

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-perpg 29005

This theorem is used by:  perpprlng  29229