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Theorem hleqnid 28698

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 2942	. . 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 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐵 ∈ 𝑃)
10		hlln.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐺 ∈ TarskiG)
12		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴(𝐾‘𝐴)𝐵)
13	3, 4, 5, 7, 9, 7, 11, 12	hlne1 28695	. 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 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5100 ‘cfv 6502 Basecbs 17150 TarskiGcstrkg 28516 Itvcitv 28522 hlGchlg 28690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7373 df-hlg 28691

This theorem is referenced by: mirbtwnhl 28770

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