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Theorem hleqnid 26699

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 2949	. . 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 484	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐵 ∈ 𝑃)
10		hlln.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐺 ∈ TarskiG)
12		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴(𝐾‘𝐴)𝐵)
13	3, 4, 5, 7, 9, 7, 11, 12	hlne1 26696	. 2 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ≠ 𝐴)
14	2, 13	mtand 816	1 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 class class class wbr 5053 ‘cfv 6380 Basecbs 16760 TarskiGcstrkg 26521 Itvcitv 26527 hlGchlg 26691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-hlg 26692

This theorem is referenced by: mirbtwnhl 26771

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