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Theorem hleqnid 29101

Description: The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-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)

**Assertion**
Ref	Expression
hleqnid	⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵)

**Proof of Theorem hleqnid**
Step	Hyp	Ref	Expression
1		neirr 2968	. . 3 ⊢ ¬ 𝐴 ≠ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → ¬ 𝐴 ≠ 𝐴)
3		ishlg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
4		ishlg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		ishlg.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
6		ishlg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐵 ∈ 𝑃)
10		hlln.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐺 ∈ TarskiG)
12		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴(𝐾‘𝐴)𝐵)
13	3, 4, 5, 7, 9, 7, 11, 12	hlne1 29098	. 2 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ≠ 𝐴)
14	2, 13	mtand 828	1 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 ‘cfv 6555 Basecbs 17413 TarskiGcstrkg 28916 Itvcitv 28922 hlGchlg 29090

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-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

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-reu 3370 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 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-hlg 29091

This theorem is used by: mirbtwnhl 29179

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