Theorem frgra2v 26526

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.)

Assertion

Ref	Expression
frgra2v	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)

Proof of Theorem frgra2v

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

Step	Hyp	Ref	Expression
1		neirr 2791	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝐴 ≠ 𝐴
2		usgraedgrn 25910	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 ≠ 𝐴)
3	2	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → 𝐴 ≠ 𝐴))
4	1, 3	mtoi 189	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐴, 𝐴} ∈ ran 𝐸)
5	4	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐴, 𝐴} ∈ ran 𝐸)
6	5	intnanrd 954	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
7		prex 4836	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐴} ∈ V
8		prex 4836	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐵} ∈ V
9	7, 8	prss 4291	. . . . . . . . . . . . 13 ⊢ (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
10	6, 9	sylnib 317	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
11		neirr 2791	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝐵 ≠ 𝐵
12		usgraedgrn 25910	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐴, 𝐵} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 ≠ 𝐵)
13	12	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ({𝐵, 𝐵} ∈ ran 𝐸 → 𝐵 ≠ 𝐵))
14	11, 13	mtoi 189	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐵, 𝐵} ∈ ran 𝐸)
15	14	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐵, 𝐵} ∈ ran 𝐸)
16	15	intnand 953	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸))
17		prex 4836	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐴} ∈ V
18		prex 4836	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐵} ∈ V
19	17, 18	prss 4291	. . . . . . . . . . . . 13 ⊢ (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
20	16, 19	sylnib 317	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
21		ioran 510	. . . . . . . . . . . 12 ⊢ (¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸) ↔ (¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∧ ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
22	10, 20, 21	sylanbrc 695	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
23		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
24		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
25	23, 24	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
26	25	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
27		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
28		preq1 4212	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
29	27, 28	preq12d 4220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
30	29	sseq1d 3595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
31	26, 30	rexprg 4182	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)))
32	31	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)))
33	22, 32	mtbird 314	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
34		reurex 3137	. . . . . . . . . 10 ⊢ (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
35	33, 34	nsyl 134	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
36	35	orcd 406	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
37		rexnal 2978	. . . . . . . . . . . 12 ⊢ (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸)
38	37	bicomi 213	. . . . . . . . . . 11 ⊢ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸)
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
40		difprsn1 4271	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
41	40	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
42	41	rexeqdv 3122	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
43		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
44	43	preq2d 4219	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
45	44	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
46	45	reubidv 3103	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
47	46	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
48	47	rexsng 4166	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑌 → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
49	48	ad3antlr 763	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
50	39, 42, 49	3bitrd 293	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
51		rexnal 2978	. . . . . . . . . . . 12 ⊢ (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)
52	51	bicomi 213	. . . . . . . . . . 11 ⊢ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)
53	52	a1i 11	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
54		difprsn2 4272	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
55	54	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
56	55	rexeqdv 3122	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
57		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
58	57	preq2d 4219	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
59	58	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
60	59	reubidv 3103	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
61	60	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
62	61	rexsng 4166	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
63	62	ad3antrrr 762	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
64	53, 56, 63	3bitrd 293	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
65	50, 64	orbi12d 742	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ((¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸)))
66	36, 65	mpbird 246	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
67		sneq 4135	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
68	67	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
69		preq2 4213	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
70	69	preq1d 4218	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
71	70	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
72	71	reubidv 3103	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
73	68, 72	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
74	73	notbid 307	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
75		sneq 4135	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → {𝑘} = {𝐵})
76	75	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
77		preq2 4213	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
78	77	preq1d 4218	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
79	78	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
80	79	reubidv 3103	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
81	76, 80	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
82	81	notbid 307	. . . . . . . . 9 ⊢ (𝑘 = 𝐵 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
83	74, 82	rexprg 4182	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
84	83	ad2antrr 758	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
85	66, 84	mpbird 246	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
86		rexnal 2978	. . . . . 6 ⊢ (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
87	85, 86	sylib 207	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
88	87	intnand 953	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
89		usgrav 25867	. . . . . 6 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
90	89	adantl 481	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
91		isfrgra 26517	. . . . 5 ⊢ (({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V) → ({𝐴, 𝐵} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
92	90, 91	syl 17	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
93	88, 92	mtbird 314	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
94	93	expcom 450	. 2 ⊢ ({𝐴, 𝐵} USGrph 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸))
95		frisusgra 26519	. . . 4 ⊢ ({𝐴, 𝐵} FriendGrph 𝐸 → {𝐴, 𝐵} USGrph 𝐸)
96	95	con3i 149	. . 3 ⊢ (¬ {𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
97	96	a1d 25	. 2 ⊢ (¬ {𝐴, 𝐵} USGrph 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸))
98	94, 97	pm2.61i 175	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-frgra 26516

This theorem is referenced by:  1to2vfriswmgra  26533  frgraregord013  26645