Theorem nfrgr2v 27136

Description: Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.) (Revised by AV, 29-Mar-2021.)

Assertion

Ref	Expression
nfrgr2v	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Proof of Theorem nfrgr2v

Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neirr 2803	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐴 ≠ 𝐴
2		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	usgredgne 26098	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ (Edg‘𝐺)) → 𝐴 ≠ 𝐴)
4	3	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝐴} ∈ (Edg‘𝐺) → 𝐴 ≠ 𝐴))
5	1, 4	mtoi 190	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
6	5	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
7	6	intnanrd 963	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)))
8		prex 4909	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐴} ∈ V
9		prex 4909	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐵} ∈ V
10	8, 9	prss 4351	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
11	7, 10	sylnib 318	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
12		neirr 2803	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐵 ≠ 𝐵
13	2	usgredgne 26098	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) → 𝐵 ≠ 𝐵)
14	13	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐵} ∈ (Edg‘𝐺) → 𝐵 ≠ 𝐵))
15	12, 14	mtoi 190	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
16	15	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
17	16	intnand 962	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)))
18		prex 4909	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐴} ∈ V
19		prex 4909	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐵} ∈ V
20	18, 19	prss 4351	. . . . . . . . . . . . . . 15 ⊢ (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
21	17, 20	sylnib 318	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
22		ioran 511	. . . . . . . . . . . . . 14 ⊢ (¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)) ↔ (¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∧ ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
23	11, 21, 22	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
24		preq1 4268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
25		preq1 4268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
26	24, 25	preq12d 4276	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
27	26	sseq1d 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺)))
28		preq1 4268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
29		preq1 4268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
30	28, 29	preq12d 4276	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
31	30	sseq1d 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
32	27, 31	rexprg 4235	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
33	32	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
34	33	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
35	23, 34	mtbird 315	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
36		reurex 3160	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
37	35, 36	nsyl 135	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
38	37	orcd 407	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
39		rexnal 2995	. . . . . . . . . . . . . 14 ⊢ (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
40	39	bicomi 214	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
41	40	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
42		difprsn1 4330	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
43	42	3ad2ant3 1084	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
44	43	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
45	44	rexeqdv 3145	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
46		preq2 4269	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
47	46	preq2d 4275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
48	47	sseq1d 3632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
49	48	reubidv 3126	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
50	49	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
51	50	rexsng 4219	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑌 → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
52	51	3ad2ant2 1083	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
53	52	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
54	41, 45, 53	3bitrd 294	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
55		rexnal 2995	. . . . . . . . . . . . . 14 ⊢ (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
56	55	bicomi 214	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
58		difprsn2 4331	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
59	58	3ad2ant3 1084	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
60	59	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
61	60	rexeqdv 3145	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
62		preq2 4269	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
63	62	preq2d 4275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
64	63	sseq1d 3632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
65	64	reubidv 3126	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
66	65	notbid 308	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
67	66	rexsng 4219	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
68	67	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
69	68	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
70	57, 61, 69	3bitrd 294	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
71	54, 70	orbi12d 746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ((¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺))))
72	38, 71	mpbird 247	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
73		sneq 4187	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
74	73	difeq2d 3728	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
75		preq2 4269	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
76	75	preq1d 4274	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
77	76	sseq1d 3632	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
78	77	reubidv 3126	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
79	74, 78	raleqbidv 3152	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
80	79	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
81		sneq 4187	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → {𝑘} = {𝐵})
82	81	difeq2d 3728	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
83		preq2 4269	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
84	83	preq1d 4274	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
85	84	sseq1d 3632	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
86	85	reubidv 3126	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
87	82, 86	raleqbidv 3152	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
88	87	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
89	80, 88	rexprg 4235	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
90	89	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
91	90	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
92	72, 91	mpbird 247	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
93		rexnal 2995	. . . . . . . 8 ⊢ (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
94	92, 93	sylib 208	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
95	94	intnand 962	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
96	95	adantlr 751	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
97		id 22	. . . . . . . . . 10 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (Vtx‘𝐺) = {𝐴, 𝐵})
98		difeq1 3721	. . . . . . . . . . 11 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝑘}))
99		reueq1 3140	. . . . . . . . . . 11 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
100	98, 99	raleqbidv 3152	. . . . . . . . . 10 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
101	97, 100	raleqbidv 3152	. . . . . . . . 9 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
102	101	anbi2d 740	. . . . . . . 8 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → ((𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
103	102	notbid 308	. . . . . . 7 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
104	103	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
105	104	adantr 481	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
106	96, 105	mpbird 247	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
107		df-nel 2898	. . . . 5 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
108		eqid 2622	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
109	108, 2	frgrusgrfrcond 27123	. . . . 5 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
110	107, 109	xchbinx 324	. . . 4 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
111	106, 110	sylibr 224	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → 𝐺 ∉ FriendGraph )
112	111	expcom 451	. 2 ⊢ (𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
113		frgrusgr 27124	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
114	113	con3i 150	. . . 4 ⊢ (¬ 𝐺 ∈ USGraph → ¬ 𝐺 ∈ FriendGraph )
115	114, 107	sylibr 224	. . 3 ⊢ (¬ 𝐺 ∈ USGraph → 𝐺 ∉ FriendGraph )
116	115	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
117	112, 116	pm2.61i 176	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-umgr 25978  df-usgr 26046  df-frgr 27121

This theorem is referenced by:  1to2vfriswmgr  27143  frgrregord013  27253