Theorem 4cycl2vnunb 30891

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 30890	. 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 4694	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → {𝐴, 𝑥} = {𝐴, 𝐵})
3		preq1 4693	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐶} = {𝐵, 𝐶})
4	2, 3	preq12d 4701	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝐵}, {𝐵, 𝐶}})
5	4	sseq1d 3939	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸))
6	5	anbi1d 643	. . . . . . . . . . . . 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 3017	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑥 ≠ 𝑦 ↔ 𝐵 ≠ 𝑦))
8	6, 7	anbi12d 644	. . . . . . . . . . . 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 4694	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐷 → {𝐴, 𝑦} = {𝐴, 𝐷})
10		preq1 4693	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐷 → {𝑦, 𝐶} = {𝐷, 𝐶})
11	9, 10	preq12d 4701	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐷 → {{𝐴, 𝑦}, {𝑦, 𝐶}} = {{𝐴, 𝐷}, {𝐷, 𝐶}})
12	11	sseq1d 3939	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐷 → ({{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸))
13	12	anbi2d 642	. . . . . . . . . . . . 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 3018	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐷 → (𝐵 ≠ 𝑦 ↔ 𝐵 ≠ 𝐷))
15	13, 14	anbi12d 644	. . . . . . . . . . . 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 3588	. . . . . . . . . . 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 1136	. . . . . . . . . 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 419	. . . . . . . . 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 418	. . . . . . . 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 89	. . . . . . 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 1135	. . . . . 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 413	. . . . 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 3114	. . . . . 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 3114	. . . . . . . 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 409	. . . . . . . . . 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 2956	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
27	26	bicomi 227	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
28	27	anbi2i 635	. . . . . . . . . 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 280	. . . . . . . . 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 3109	. . . . . . . 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 280	. . . . . . 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 3109	. . . . . 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 280	. . . . 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 237	. . . 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 494	. . 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 4694	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝐴, 𝑥} = {𝐴, 𝑦})
37		preq1 4693	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥, 𝐶} = {𝑦, 𝐶})
38	36, 37	preq12d 4701	. . . . 5 ⊢ (𝑥 = 𝑦 → {{𝐴, 𝑥}, {𝑥, 𝐶}} = {{𝐴, 𝑦}, {𝑦, 𝐶}})
39	38	sseq1d 3939	. . . 4 ⊢ (𝑥 = 𝑦 → ({{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑦}, {𝑦, 𝐶}} ⊆ 𝐸))
40	39	reu4 3688	. . 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 332	. 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 1808	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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363   ⊆ wss 3895  {cpr 4585

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-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-v 3452  df-ss 3896  df-un 4151  df-sn 4584  df-pr 4586

This theorem is used by:  n4cyclfrgr  30892  4cyclusnfrgr  30893  pgn4cyclex  49224