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Theorem hleqnid 29074

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 2964	. . 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 29071	. 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 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6538 Basecbs 17387 TarskiGcstrkg 28889 Itvcitv 28895 hlGchlg 29063

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-hlg 29064

This theorem is used by: mirbtwnhl 29152

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