Theorem 4cycl2vnunb 28654

Description: In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.)

Assertion

Ref	Expression
4cycl2vnunb	⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐸   𝑥,𝑉

Allowed substitution hint:   𝐷(𝑥)

Proof of Theorem 4cycl2vnunb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4cycl2v2nb 28653	. 2 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
2		preq2 4670	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → {𝐴, 𝑥} = {𝐴, 𝐵})
3		preq1 4669	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐶} = {𝐵, 𝐶})
4	2, 3	preq12d 4677	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝐵}, {𝐵, 𝐶}})
5	4	sseq1d 3952	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸))
6	5	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ↔ ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸)))
7		neeq1 3006	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ≠ 𝑦 ↔ 𝐵 ≠ 𝑦))
8	6, 7	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦) ↔ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝑦)))
9		preq2 4670	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐷 → {𝐴, 𝑦} = {𝐴, 𝐷})
10		preq1 4669	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐷 → {𝑦, 𝐶} = {𝐷, 𝐶})
11	9, 10	preq12d 4677	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐷 → {{𝐴, 𝑦}, {𝑦, 𝐶}} = {{𝐴, 𝐷}, {𝐷, 𝐶}})
12	11	sseq1d 3952	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐷 → ({{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
13	12	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐷 → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ↔ ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)))
14		neeq2 3007	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐷 → (𝐵 ≠ 𝑦 ↔ 𝐵 ≠ 𝐷))
15	13, 14	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐷 → ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝑦) ↔ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝐷)))
16	8, 15	rspc2ev 3572	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝐷)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
17	16	3expa 1117	. . . . . . . . . 10 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) ∧ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝐷)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
18	17	expcom 414	. . . . . . . . 9 ⊢ ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ 𝐵 ≠ 𝐷) → ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦)))
19	18	ex 413	. . . . . . . 8 ⊢ (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → (𝐵 ≠ 𝐷 → ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))))
20	19	com13 88	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 ≠ 𝐷 → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))))
21	20	3impia 1116	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦)))
22	21	impcom 408	. . . . 5 ⊢ ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
23		rexnal 3169	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ¬ ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ¬ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
24		rexnal 3169	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑉 ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ¬ ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
25		annim 404	. . . . . . . . . 10 ⊢ ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ ¬ 𝑥 = 𝑦) ↔ ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
26		df-ne 2944	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
27	26	bicomi 223	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
28	27	anbi2i 623	. . . . . . . . . 10 ⊢ ((({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ ¬ 𝑥 = 𝑦) ↔ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
29	25, 28	bitr3i 276	. . . . . . . . 9 ⊢ (¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
30	29	rexbii 3181	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑉 ¬ (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
31	24, 30	bitr3i 276	. . . . . . 7 ⊢ (¬ ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
32	31	rexbii 3181	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ¬ ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
33	23, 32	bitr3i 276	. . . . 5 ⊢ (¬ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) ∧ 𝑥 ≠ 𝑦))
34	22, 33	sylibr 233	. . . 4 ⊢ ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦))
35	34	intnand 489	. . 3 ⊢ ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ (∃𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦)))
36		preq2 4670	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝐴, 𝑥} = {𝐴, 𝑦})
37		preq1 4669	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥, 𝐶} = {𝑦, 𝐶})
38	36, 37	preq12d 4677	. . . . 5 ⊢ (𝑥 = 𝑦 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝑦}, {𝑦, 𝐶}})
39	38	sseq1d 3952	. . . 4 ⊢ (𝑥 = 𝑦 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸))
40	39	reu4 3666	. . 3 ⊢ (∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ (∃𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸) → 𝑥 = 𝑦)))
41	35, 40	sylnibr 329	. 2 ⊢ ((({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)
42	1, 41	stoic3 1779	1 ⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066   ⊆ wss 3887  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564

This theorem is referenced by:  n4cyclfrgr  28655  4cyclusnfrgr  28656