Theorem btwnlng13 35002

Description: If 𝑍 is between 𝑋 and 𝑌, or 𝑌 is between 𝑋 and 𝑍, then 𝑍 lies on the line 𝑋𝑌. (Contributed by SS, 4-Jun-2026.)

Hypotheses

Ref	Expression
btwnlng13.p	⊢ 𝑃 = (Base‘𝐺)
btwnlng13.i	⊢ 𝐼 = (Itv‘𝐺)
btwnlng13.l	⊢ 𝐿 = (LineG‘𝐺)
btwnlng13.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
btwnlng13.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
btwnlng13.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
btwnlng13.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwnlng13.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)
btwnlng13.1	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

Assertion

Ref	Expression
btwnlng13	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng13

Step	Hyp	Ref	Expression
1		btwnlng13.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		btwnlng13.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		btwnlng13.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
4		btwnlng13.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
6		btwnlng13.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
8		btwnlng13.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
10		btwnlng13.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
12		btwnlng13.d	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ≠ 𝑌)
14		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
15	1, 2, 3, 5, 7, 9, 11, 13, 14	btwnlng1 28854	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐿𝑌))
16	4	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG)
17	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃)
18	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃)
19	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃)
20	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ≠ 𝑌)
21		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍))
22	1, 2, 3, 16, 17, 18, 19, 20, 21	btwnlng3 28856	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ (𝑋𝐿𝑌))
23		btwnlng13.1	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
24	15, 22, 23	mpjaodan 973	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  TarskiGcstrkg 28662  Itvcitv 28668  LineGclng 28669

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 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  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-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-trkg 28688

This theorem is referenced by:  morleylemrneab  35003