Theorem nfrgr2v 30880

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 2964	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐴 ≠ 𝐴
2		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	2	usgredgne 29794	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ (Edg‘𝐺)) → 𝐴 ≠ 𝐴)
4	3	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐴, 𝐴} ∈ (Edg‘𝐺) → 𝐴 ≠ 𝐴))
5	1, 4	mtoi 202	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
6	5	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
7	6	intnanrd 495	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)))
8		prex 5395	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐴} ∈ V
9		prex 5395	. . . . . . . . . . . . . . . 16 ⊢ {𝐴, 𝐵} ∈ V
10	8, 9	prss 4775	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
11	7, 10	sylnib 331	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
12		neirr 2964	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 𝐵 ≠ 𝐵
13	2	usgredgne 29794	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) → 𝐵 ≠ 𝐵)
14	13	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → ({𝐵, 𝐵} ∈ (Edg‘𝐺) → 𝐵 ≠ 𝐵))
15	12, 14	mtoi 202	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
16	15	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
17	16	intnand 494	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)))
18		prex 5395	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐴} ∈ V
19		prex 5395	. . . . . . . . . . . . . . . 16 ⊢ {𝐵, 𝐵} ∈ V
20	18, 19	prss 4775	. . . . . . . . . . . . . . 15 ⊢ (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
21	17, 20	sylnib 331	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
22		ioran 999	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
24		preq1 4690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
25		preq1 4690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
26	24, 25	preq12d 4698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
27	26	sseq1d 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺)))
28		preq1 4690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
29		preq1 4690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
30	28, 29	preq12d 4698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
31	30	sseq1d 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
32	27, 31	rexprg 4654	. . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 328	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
36		reurex 3369	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
37	35, 36	nsyl 141	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
38	37	orcd 887	. . . . . . . . . 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 3114	. . . . . . . . . . . . . 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 227	. . . . . . . . . . . . 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 4759	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
43	42	3ad2ant3 1153	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
44	43	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
45	44	rexeqdv 3320	. . . . . . . . . . . 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 4691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
47	46	preq2d 4697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
48	47	sseq1d 3940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
49	48	reubidv 3381	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
50	49	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
51	50	rexsng 4633	. . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 308	. . . . . . . . . . 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 3114	. . . . . . . . . . . . . 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 227	. . . . . . . . . . . . 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 4760	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
59	58	3ad2ant3 1153	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
60	59	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
61	60	rexeqdv 3320	. . . . . . . . . . . 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 4691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
63	62	preq2d 4697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
64	63	sseq1d 3940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
65	64	reubidv 3381	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
66	65	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
67	66	rexsng 4633	. . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 308	. . . . . . . . . . 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 932	. . . . . . . . . 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 260	. . . . . . . . 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 4590	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
74	73	difeq2d 4062	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
75		preq2 4691	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
76	75	preq1d 4696	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
77	76	sseq1d 3940	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
78	77	reubidv 3381	. . . . . . . . . . . . . 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 3334	. . . . . . . . . . . . 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 321	. . . . . . . . . . . 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 4590	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → {𝑘} = {𝐵})
82	81	difeq2d 4062	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
83		preq2 4691	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
84	83	preq1d 4696	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
85	84	sseq1d 3940	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
86	85	reubidv 3381	. . . . . . . . . . . . . 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 3334	. . . . . . . . . . . . 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 321	. . . . . . . . . . . 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 4654	. . . . . . . . . . 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 1150	. . . . . . . . . 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 486	. . . . . . . . 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 260	. . . . . . . 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 3114	. . . . . . . 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 221	. . . . . . 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 494	. . . . . 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 728	. . . . 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 23	. . . . . . . . . 10 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → (Vtx‘𝐺) = {𝐴, 𝐵})
98		difeq1 4055	. . . . . . . . . . 11 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝑘}))
99		reueq1 3397	. . . . . . . . . . 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 3334	. . . . . . . . . 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 3334	. . . . . . . . 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 642	. . . . . . . 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 321	. . . . . . 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 487	. . . . . 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 486	. . . . 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 260	. . . 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 3062	. . . . 5 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
108		eqid 2760	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
109	108, 2	isfrgr 30868	. . . . 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 337	. . . 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 237	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph) → 𝐺 ∉ FriendGraph )
112	111	expcom 419	. 2 ⊢ (𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
113		frgrusgr 30869	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
114	113	con3i 155	. . . 4 ⊢ (¬ 𝐺 ∈ USGraph → ¬ 𝐺 ∈ FriendGraph )
115	114, 107	sylibr 237	. . 3 ⊢ (¬ 𝐺 ∈ USGraph → 𝐺 ∉ FriendGraph )
116	115	a1d 26	. 2 ⊢ (¬ 𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
117	112, 116	pm2.61i 184	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∉ wnel 3061  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363   ⊆ wss 3896   ∖ cdif 4044  {csn 4584  {cpr 4586  ‘cfv 6541  Vtxcvtx 29581  Edgcedg 29632  USGraphcusgr 29737   FriendGraph cfrgr 30866

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-oadd 8485  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-dju 9987  df-card 10025  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-n0 12612  df-z 12699  df-uz 12971  df-fz 13645  df-hash 14480  df-edg 29633  df-umgr 29668  df-usgr 29739  df-frgr 30867

This theorem is used by:  1to2vfriswmgr  30887  frgrregord013  31003