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Theorem axtgupdim2 28230

Description: Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. Three points 𝑋, 𝑌 and 𝑍 equidistant to two given two points 𝑈 and 𝑉 must be colinear. (Contributed by Thierry Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.)

**Hypotheses**
Ref	Expression
axtrkge.p	⊢ 𝑃 = (Base‘𝐺)
axtrkge.d	⊢ − = (dist‘𝐺)
axtrkge.i	⊢ 𝐼 = (Itv‘𝐺)
axtgupdim2.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgupdim2.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
axtgupdim2.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
axtgupdim2.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
axtgupdim2.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
axtgupdim2.0	⊢ (𝜑 → 𝑈 ≠ 𝑉)
axtgupdim2.1	⊢ (𝜑 → (𝑈 − 𝑋) = (𝑉 − 𝑋))
axtgupdim2.2	⊢ (𝜑 → (𝑈 − 𝑌) = (𝑉 − 𝑌))
axtgupdim2.3	⊢ (𝜑 → (𝑈 − 𝑍) = (𝑉 − 𝑍))
axtgupdim2.w	⊢ (𝜑 → 𝐺 ∈ 𝑉)
axtgupdim2.g	⊢ (𝜑 → ¬ 𝐺Dim_TarskiG≥3)

**Assertion**
Ref	Expression
axtgupdim2	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

Proof of Theorem axtgupdim2 Dummy variables 𝑣 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axtgupdim2.1	. . 3 ⊢ (𝜑 → (𝑈 − 𝑋) = (𝑉 − 𝑋))
2		axtgupdim2.2	. . 3 ⊢ (𝜑 → (𝑈 − 𝑌) = (𝑉 − 𝑌))
3		axtgupdim2.3	. . 3 ⊢ (𝜑 → (𝑈 − 𝑍) = (𝑉 − 𝑍))
4		axtgupdim2.0	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
5		axtgupdim2.g	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐺Dim_TarskiG≥3)
6		axtgupdim2.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		axtrkge.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Base‘𝐺)
8		axtrkge.d	. . . . . . . . . . . . 13 ⊢ − = (dist‘𝐺)
9		axtrkge.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (Itv‘𝐺)
10	7, 8, 9	istrkg3ld 28220	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝑉 → (𝐺Dim_TarskiG≥3 ↔ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
11	6, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺Dim_TarskiG≥3 ↔ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
12	5, 11	mtbid 324	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
13		ralnex2 3127	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ¬ (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ ∃𝑢 ∈ 𝑃 ∃𝑣 ∈ 𝑃 (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
14	12, 13	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ¬ (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
15		axtgupdim2.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
16		axtgupdim2.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
17		neeq1 2997	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑈 → (𝑢 ≠ 𝑣 ↔ 𝑈 ≠ 𝑣))
18		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑈 → (𝑢 − 𝑥) = (𝑈 − 𝑥))
19	18	eqeq1d 2728	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑈 → ((𝑢 − 𝑥) = (𝑣 − 𝑥) ↔ (𝑈 − 𝑥) = (𝑣 − 𝑥)))
20		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑈 → (𝑢 − 𝑦) = (𝑈 − 𝑦))
21	20	eqeq1d 2728	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑈 → ((𝑢 − 𝑦) = (𝑣 − 𝑦) ↔ (𝑈 − 𝑦) = (𝑣 − 𝑦)))
22		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = 𝑈 → (𝑢 − 𝑧) = (𝑈 − 𝑧))
23	22	eqeq1d 2728	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑈 → ((𝑢 − 𝑧) = (𝑣 − 𝑧) ↔ (𝑈 − 𝑧) = (𝑣 − 𝑧)))
24	19, 21, 23	3anbi123d 1432	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑈 → (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ↔ ((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧))))
25	24	anbi1d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑈 → ((((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
26	25	rexbidv 3172	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑈 → (∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
27	26	2rexbidv 3213	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑈 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
28	17, 27	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑈 → ((𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑈 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
29	28	notbid 318	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → (¬ (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
30		neeq2 2998	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑉 → (𝑈 ≠ 𝑣 ↔ 𝑈 ≠ 𝑉))
31		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → (𝑣 − 𝑥) = (𝑉 − 𝑥))
32	31	eqeq2d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑉 → ((𝑈 − 𝑥) = (𝑣 − 𝑥) ↔ (𝑈 − 𝑥) = (𝑉 − 𝑥)))
33		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → (𝑣 − 𝑦) = (𝑉 − 𝑦))
34	33	eqeq2d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑉 → ((𝑈 − 𝑦) = (𝑣 − 𝑦) ↔ (𝑈 − 𝑦) = (𝑉 − 𝑦)))
35		oveq1 7412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → (𝑣 − 𝑧) = (𝑉 − 𝑧))
36	35	eqeq2d 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑉 → ((𝑈 − 𝑧) = (𝑣 − 𝑧) ↔ (𝑈 − 𝑧) = (𝑉 − 𝑧)))
37	32, 34, 36	3anbi123d 1432	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑉 → (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ↔ ((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧))))
38	37	anbi1d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑉 → ((((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
39	38	rexbidv 3172	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑉 → (∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
40	39	2rexbidv 3213	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑉 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
41	30, 40	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑉 → ((𝑈 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
42	41	notbid 318	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑉 → (¬ (𝑈 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
43	29, 42	rspc2v 3617	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑃 ∧ 𝑉 ∈ 𝑃) → (∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ¬ (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) → ¬ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
44	15, 16, 43	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ¬ (𝑢 ≠ 𝑣 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑢 − 𝑥) = (𝑣 − 𝑥) ∧ (𝑢 − 𝑦) = (𝑣 − 𝑦) ∧ (𝑢 − 𝑧) = (𝑣 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) → ¬ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
45	14, 44	mpd 15	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
46		imnan 399	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑉 → ¬ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈 ≠ 𝑉 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
47	45, 46	sylibr 233	. . . . . . 7 ⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
48	4, 47	mpd 15	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
49		ralnex3 3128	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ¬ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ¬ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
50	48, 49	sylibr 233	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ¬ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
51		axtgupdim2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
52		axtgupdim2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
53		axtgupdim2.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
54		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑈 − 𝑥) = (𝑈 − 𝑋))
55		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑉 − 𝑥) = (𝑉 − 𝑋))
56	54, 55	eqeq12d 2742	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑈 − 𝑥) = (𝑉 − 𝑥) ↔ (𝑈 − 𝑋) = (𝑉 − 𝑋)))
57	56	3anbi1d 1436	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ↔ ((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧))))
58		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
59	58	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
60		eleq1 2815	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
61		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
62	61	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
63	59, 60, 62	3orbi123d 1431	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
64	63	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
65	57, 64	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
66	65	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
67		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑈 − 𝑦) = (𝑈 − 𝑌))
68		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑉 − 𝑦) = (𝑉 − 𝑌))
69	67, 68	eqeq12d 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((𝑈 − 𝑦) = (𝑉 − 𝑦) ↔ (𝑈 − 𝑌) = (𝑉 − 𝑌)))
70	69	3anbi2d 1437	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ↔ ((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧))))
71		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
72	71	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
73		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
74	73	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
75		eleq1 2815	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
76	72, 74, 75	3orbi123d 1431	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
77	76	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
78	70, 77	anbi12d 630	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
79	78	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
80		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (𝑈 − 𝑧) = (𝑈 − 𝑍))
81		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (𝑉 − 𝑧) = (𝑉 − 𝑍))
82	80, 81	eqeq12d 2742	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → ((𝑈 − 𝑧) = (𝑉 − 𝑧) ↔ (𝑈 − 𝑍) = (𝑉 − 𝑍)))
83	82	3anbi3d 1438	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ↔ ((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍))))
84		eleq1 2815	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
85		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
86	85	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
87		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
88	87	eleq2d 2813	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
89	84, 86, 88	3orbi123d 1431	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
90	89	notbid 318	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
91	83, 90	anbi12d 630	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → ((((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
92	91	notbid 318	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
93	66, 79, 92	rspc3v 3622	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ¬ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
94	51, 52, 53, 93	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ¬ (((𝑈 − 𝑥) = (𝑉 − 𝑥) ∧ (𝑈 − 𝑦) = (𝑉 − 𝑦) ∧ (𝑈 − 𝑧) = (𝑉 − 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
95	50, 94	mpd 15	. . . 4 ⊢ (𝜑 → ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
96		imnan 399	. . . 4 ⊢ ((((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) ↔ ¬ (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
97	95, 96	sylibr 233	. . 3 ⊢ (𝜑 → (((𝑈 − 𝑋) = (𝑉 − 𝑋) ∧ (𝑈 − 𝑌) = (𝑉 − 𝑌) ∧ (𝑈 − 𝑍) = (𝑉 − 𝑍)) → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
98	1, 2, 3, 97	mp3and 1460	. 2 ⊢ (𝜑 → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
99	98	notnotrd 133	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 3c3 12272 Basecbs 17153 distcds 17215 Dim_TarskiG≥cstrkgld 28190 Itvcitv 28192

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-trkgld 28211

This theorem is referenced by: (None)

W3C validator