Theorem perpin 29257

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 4290	. . 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 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 29242	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (LineG‘𝐺))
10	7, 8, 3	perpln2 29243	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ran (LineG‘𝐺))
11	4, 5, 6, 7, 8, 9, 10	isperp 29244	. . 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 3174	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ∩ cin 4092  ∅c0 4281   class class class wbr 5108  ‘cfv 6566  ⟨“cs3 15035  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952  LineGclng 28953  ∟Gcrag 29225  ⟂Gcperpg 29227

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-perpg 29228

This theorem is used by:  perpprlng  29485