Theorem perpin 29193

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 4287	. . 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 2761	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
5		eqid 2761	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
6		eqid 2761	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
7		eqid 2761	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
8		perpin.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	7, 8, 3	perpln1 29178	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran (LineG‘𝐺))
10	7, 8, 3	perpln2 29179	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ran (LineG‘𝐺))
11	4, 5, 6, 7, 8, 9, 10	isperp 29180	. . 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 3171	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∩ cin 3898  ∅c0 4279   class class class wbr 5103  ‘cfv 6537  ⟨“cs3 14986  Basecbs 17380  distcds 17430  TarskiGcstrkg 28882  Itvcitv 28888  LineGclng 28889  ∟Gcrag 29161  ⟂Gcperpg 29163

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fv 6545  df-perpg 29164

This theorem is used by:  perpprlng  29421