Theorem nfrgr2v 30081

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 2946	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐴 ≠ 𝐴
2		eqid 2728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	usgredgne 29018	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ (Edg‘𝐺)) → 𝐴 ≠ 𝐴)
4	3	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝐴} ∈ (Edg‘𝐺) → 𝐴 ≠ 𝐴))
5	1, 4	mtoi 198	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
6	5	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
7	6	intnanrd 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)))
8		prex 5434	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐴} ∈ V
9		prex 5434	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐵} ∈ V
10	8, 9	prss 4824	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
11	7, 10	sylnib 328	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
12		neirr 2946	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐵 ≠ 𝐵
13	2	usgredgne 29018	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) → 𝐵 ≠ 𝐵)
14	13	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐵} ∈ (Edg‘𝐺) → 𝐵 ≠ 𝐵))
15	12, 14	mtoi 198	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
16	15	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
17	16	intnand 488	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)))
18		prex 5434	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐴} ∈ V
19		prex 5434	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐵} ∈ V
20	18, 19	prss 4824	. . . . . . . . . . . . . . 15 ⊢ (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
21	17, 20	sylnib 328	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
22		ioran 982	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
24		preq1 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
25		preq1 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
26	24, 25	preq12d 4746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
27	26	sseq1d 4011	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺)))
28		preq1 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
29		preq1 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
30	28, 29	preq12d 4746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
31	30	sseq1d 4011	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
32	27, 31	rexprg 4701	. . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 325	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
36		reurex 3377	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
37	35, 36	nsyl 140	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
38	37	orcd 872	. . . . . . . . . 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 3097	. . . . . . . . . . . . . 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 223	. . . . . . . . . . . . 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 4804	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
43	42	3ad2ant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
44	43	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
45	44	rexeqdv 3323	. . . . . . . . . . . 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 4739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
47	46	preq2d 4745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
48	47	sseq1d 4011	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
49	48	reubidv 3391	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
50	49	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
51	50	rexsng 4679	. . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 305	. . . . . . . . . . 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 3097	. . . . . . . . . . . . . 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 223	. . . . . . . . . . . . 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 4805	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
59	58	3ad2ant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
60	59	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
61	60	rexeqdv 3323	. . . . . . . . . . . 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 4739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
63	62	preq2d 4745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
64	63	sseq1d 4011	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
65	64	reubidv 3391	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
66	65	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
67	66	rexsng 4679	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 305	. . . . . . . . . . 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 917	. . . . . . . . . 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 257	. . . . . . . . 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 4639	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
74	73	difeq2d 4120	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
75		preq2 4739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
76	75	preq1d 4744	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
77	76	sseq1d 4011	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
78	77	reubidv 3391	. . . . . . . . . . . . . 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 3339	. . . . . . . . . . . . 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 318	. . . . . . . . . . . 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 4639	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → {𝑘} = {𝐵})
82	81	difeq2d 4120	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
83		preq2 4739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
84	83	preq1d 4744	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
85	84	sseq1d 4011	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
86	85	reubidv 3391	. . . . . . . . . . . . . 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 3339	. . . . . . . . . . . . 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 318	. . . . . . . . . . . 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 4701	. . . . . . . . . . 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 1130	. . . . . . . . . 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 480	. . . . . . . . 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 257	. . . . . . . 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 3097	. . . . . . . 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 217	. . . . . . 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 488	. . . . . 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 714	. . . . 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 4113	. . . . . . . . . . 11 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝑘}))
99		reueq1 3412	. . . . . . . . . . 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 3339	. . . . . . . . . 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 3339	. . . . . . . . 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 629	. . . . . . . 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 318	. . . . . . 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 481	. . . . . 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 480	. . . . 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 257	. . . 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 3044	. . . . 5 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
108		eqid 2728	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
109	108, 2	isfrgr 30069	. . . . 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 334	. . . 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 233	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph) → 𝐺 ∉ FriendGraph )
112	111	expcom 413	. 2 ⊢ (𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
113		frgrusgr 30070	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
114	113	con3i 154	. . . 4 ⊢ (¬ 𝐺 ∈ USGraph → ¬ 𝐺 ∈ FriendGraph )
115	114, 107	sylibr 233	. . 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 182	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937   ∉ wnel 3043  ∀wral 3058  ∃wrex 3067  ∃!wreu 3371   ∖ cdif 3944   ⊆ wss 3947  {csn 4629  {cpr 4631  ‘cfv 6548  Vtxcvtx 28808  Edgcedg 28859  USGraphcusgr 28961   FriendGraph cfrgr 30067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-oadd 8490  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-hash 14322  df-edg 28860  df-umgr 28895  df-usgr 28963  df-frgr 30068

This theorem is referenced by:  1to2vfriswmgr  30088  frgrregord013  30204