btwnhl1 - Metamath Proof Explorer

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Theorem btwnhl1 28620

Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)

**Hypotheses**
Ref	Expression
ishlg.p	⊢ 𝑃 = (Base‘𝐺)
ishlg.i	⊢ 𝐼 = (Itv‘𝐺)
ishlg.k	⊢ 𝐾 = (hlG‘𝐺)
ishlg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ishlg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
ishlg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
hlln.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hltr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
btwnhl1.1	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
btwnhl1.2	⊢ (𝜑 → 𝐴 ≠ 𝐵)
btwnhl1.3	⊢ (𝜑 → 𝐶 ≠ 𝐴)

**Assertion**
Ref	Expression
btwnhl1	⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵)

**Proof of Theorem btwnhl1**
Step	Hyp	Ref	Expression
1		btwnhl1.3	. 2 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
2		btwnhl1.2	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3	2	necomd 2996	. 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
4		btwnhl1.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
5	4	orcd 874	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
6		ishlg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
7		ishlg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
8		ishlg.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
9		ishlg.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		ishlg.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11		ishlg.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12		hlln.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
13	6, 7, 8, 9, 10, 11, 12	ishlg 28610	. 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐴)𝐵 ↔ (𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐴 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))))
14	1, 3, 5, 13	mpbir3and 1343	1 ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TarskiGcstrkg 28435 Itvcitv 28441 hlGchlg 28608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-hlg 28609

This theorem is referenced by: outpasch 28763 hlpasch 28764 lnopp2hpgb 28771 dfcgra2 28838

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