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Theorem colline 27897

Description: Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.)

**Hypotheses**
Ref	Expression
tglineintmo.p	⊢ 𝑃 = (Base‘𝐺)
tglineintmo.i	⊢ 𝐼 = (Itv‘𝐺)
tglineintmo.l	⊢ 𝐿 = (LineG‘𝐺)
tglineintmo.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
colline.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
colline.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
colline.3	⊢ (𝜑 → 𝑍 ∈ 𝑃)
colline.4	⊢ (𝜑 → 2 ≤ (♯‘𝑃))

**Assertion**
Ref	Expression
colline	⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)))

Distinct variable groups: 𝐿,𝑎 𝑋,𝑎 𝑌,𝑎 𝑍,𝑎 𝜑,𝑎 Allowed substitution hints: 𝑃(𝑎) 𝐺(𝑎) 𝐼(𝑎)

Proof of Theorem colline Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineintmo.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
2		tglineintmo.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineintmo.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineintmo.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝐺 ∈ TarskiG)
6		colline.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ∈ 𝑃)
8		simplr 767	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑥 ∈ 𝑃)
9		simpr 485	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ≠ 𝑥)
10	1, 2, 3, 5, 7, 8, 9	tgelrnln 27878	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → (𝑋𝐿𝑥) ∈ ran 𝐿)
11	1, 2, 3, 5, 7, 8, 9	tglinerflx1 27881	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ∈ (𝑋𝐿𝑥))
12		simp-4r 782	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑌 = 𝑍)
13		simpllr 774	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 = 𝑍)
14	13, 11	eqeltrrd 2834	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑍 ∈ (𝑋𝐿𝑥))
15	12, 14	eqeltrd 2833	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑌 ∈ (𝑋𝐿𝑥))
16		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑋𝐿𝑥)))
17		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑋𝐿𝑥)))
18		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑋𝐿𝑥)))
19	16, 17, 18	3anbi123d 1436	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑥) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))))
20	19	rspcev 3612	. . . . . . 7 ⊢ (((𝑋𝐿𝑥) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
21	10, 11, 15, 14, 20	syl13anc 1372	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
22		eqid 2732	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
23		colline.4	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
24	1, 22, 2, 4, 23, 6	tglowdim1i 27749	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝑋 ≠ 𝑥)
25	24	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑥 ∈ 𝑃 𝑋 ≠ 𝑥)
26	21, 25	r19.29a 3162	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
27	4	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝐺 ∈ TarskiG)
28	6	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ∈ 𝑃)
29		colline.3	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
30	29	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑍 ∈ 𝑃)
31		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ≠ 𝑍)
32	1, 2, 3, 27, 28, 30, 31	tgelrnln 27878	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → (𝑋𝐿𝑍) ∈ ran 𝐿)
33	1, 2, 3, 27, 28, 30, 31	tglinerflx1 27881	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ∈ (𝑋𝐿𝑍))
34		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑌 = 𝑍)
35	1, 2, 3, 27, 28, 30, 31	tglinerflx2 27882	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑍 ∈ (𝑋𝐿𝑍))
36	34, 35	eqeltrd 2833	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑌 ∈ (𝑋𝐿𝑍))
37		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑋𝐿𝑍)))
38		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑋𝐿𝑍)))
39		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑋𝐿𝑍)))
40	37, 38, 39	3anbi123d 1436	. . . . . . 7 ⊢ (𝑎 = (𝑋𝐿𝑍) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))))
41	40	rspcev 3612	. . . . . 6 ⊢ (((𝑋𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
42	32, 33, 36, 35, 41	syl13anc 1372	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
43	26, 42	pm2.61dane 3029	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
44	43	adantlr 713	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
45		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝜑)
46		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
47	46	neneqd 2945	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → ¬ 𝑌 = 𝑍)
48		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
49		orel2 889	. . . . . 6 ⊢ (¬ 𝑌 = 𝑍 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍)))
50	47, 48, 49	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
51	4	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝐺 ∈ TarskiG)
52		colline.2	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
53	52	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑃)
54	29	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑃)
55		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
56	1, 2, 3, 51, 53, 54, 55	tgelrnln 27878	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
57	45, 50, 46, 56	syl21anc 836	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
58	1, 2, 3, 51, 53, 54, 55	tglinerflx1 27881	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
59	45, 50, 46, 58	syl21anc 836	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
60	1, 2, 3, 51, 53, 54, 55	tglinerflx2 27882	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
61	45, 50, 46, 60	syl21anc 836	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
62		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑌𝐿𝑍)))
63		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑌𝐿𝑍)))
64		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑌𝐿𝑍)))
65	62, 63, 64	3anbi123d 1436	. . . . 5 ⊢ (𝑎 = (𝑌𝐿𝑍) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))))
66	65	rspcev 3612	. . . 4 ⊢ (((𝑌𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
67	57, 50, 59, 61, 66	syl13anc 1372	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
68	44, 67	pm2.61dane 3029	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
69		df-ne 2941	. . . . . 6 ⊢ (𝑌 ≠ 𝑍 ↔ ¬ 𝑌 = 𝑍)
70		simplr1 1215	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ 𝑎)
71	4	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝐺 ∈ TarskiG)
72	52	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑃)
73	29	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑃)
74		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
75		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑎 ∈ ran 𝐿)
76		simplr2 1216	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑎)
77		simplr3 1217	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑎)
78	1, 2, 3, 71, 72, 73, 74, 74, 75, 76, 77	tglinethru 27884	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑎 = (𝑌𝐿𝑍))
79	70, 78	eleqtrd 2835	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
80	79	ex 413	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑌 ≠ 𝑍 → 𝑋 ∈ (𝑌𝐿𝑍)))
81	69, 80	biimtrrid 242	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (¬ 𝑌 = 𝑍 → 𝑋 ∈ (𝑌𝐿𝑍)))
82	81	orrd 861	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑌 = 𝑍 ∨ 𝑋 ∈ (𝑌𝐿𝑍)))
83	82	orcomd 869	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
84	83	r19.29an 3158	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
85	68, 84	impbida 799	1 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 class class class wbr 5148 ran crn 5677 ‘cfv 6543 (class class class)co 7408 ≤ cle 11248 2c2 12266 ♯chash 14289 Basecbs 17143 distcds 17205 TarskiGcstrkg 27675 Itvcitv 27681 LineGclng 27682

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-trkgc 27696 df-trkgb 27697 df-trkgcb 27698 df-trkg 27701 df-cgrg 27759

This theorem is referenced by: (None)

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