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Theorem 3vfriswmgra 26532

Description: Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

**Assertion**
Ref	Expression
3vfriswmgra	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝑉 FriendGrph 𝐸 → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))

Distinct variable groups: 𝐴,ℎ,𝑣,𝑤 𝐵,ℎ,𝑣,𝑤 𝐶,ℎ,𝑣,𝑤 ℎ,𝐸,𝑣,𝑤 ℎ,𝑉,𝑣,𝑤 𝑣,𝑋,𝑤 𝑣,𝑌,𝑤 Allowed substitution hints: 𝑋(ℎ) 𝑌(ℎ) 𝑍(𝑤,𝑣,ℎ)

**Proof of Theorem 3vfriswmgra**
Step	Hyp	Ref	Expression
1		frisusgra 26519	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → {𝐴, 𝐵, 𝐶} USGrph 𝐸)
2		frgra3v 26529	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
3	2	3adant3 1074	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
4	3	imp 444	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)))
5		prcom 4211	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
6	5	eleq1i 2679	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸)
7	6	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ ({𝐶, 𝐴} ∈ ran 𝐸 → {𝐴, 𝐶} ∈ ran 𝐸)
8	7	3ad2ant3 1077	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐴, 𝐶} ∈ ran 𝐸)
9	8	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → {𝐴, 𝐶} ∈ ran 𝐸)
10		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
11	10	3adant3 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
12	11	3ad2ant1 1075	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
13	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
14		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐵 → 𝐴 ≠ 𝐵)
15	14	3ad2ant1 1075	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐵)
16	15	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐴 ≠ 𝐵)
17	16	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 ≠ 𝐵)
18		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → {𝐴, 𝐵, 𝐶} USGrph 𝐸)
19	13, 17, 18	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸))
20		id 22	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} ∈ ran 𝐸 → {𝐴, 𝐵} ∈ ran 𝐸)
21	20	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐴, 𝐵} ∈ ran 𝐸)
22		3vfriswmgralem 26531	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
23	22	imp 444	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
24	19, 21, 23	syl2an 493	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
25	9, 24	jca 553	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
26		simpr2 1061	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → {𝐵, 𝐶} ∈ ran 𝐸)
27		pm3.22 464	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋))
28	27	3adant3 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋))
29	28	3ad2ant1 1075	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋))
30	29	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋))
31		necom 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
32	31	biimpi 205	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
33	32	3ad2ant1 1075	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐴)
34	33	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐵 ≠ 𝐴)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 ≠ 𝐴)
36		tpcoma 4229	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
37	36	breq1i 4590	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 ↔ {𝐵, 𝐴, 𝐶} USGrph 𝐸)
38	37	biimpi 205	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → {𝐵, 𝐴, 𝐶} USGrph 𝐸)
39	38	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → {𝐵, 𝐴, 𝐶} USGrph 𝐸)
40	30, 35, 39	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋) ∧ 𝐵 ≠ 𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸))
41		prcom 4211	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
42	41	eleq1i 2679	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸)
43	42	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} ∈ ran 𝐸 → {𝐵, 𝐴} ∈ ran 𝐸)
44	43	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → {𝐵, 𝐴} ∈ ran 𝐸)
45		3vfriswmgralem 26531	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋) ∧ 𝐵 ≠ 𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) → ({𝐵, 𝐴} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸))
46	45	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋) ∧ 𝐵 ≠ 𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸)
47		reueq1 3117	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} = {𝐵, 𝐴} → (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸))
48	41, 47	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ ran 𝐸)
49	46, 48	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋) ∧ 𝐵 ≠ 𝐴 ∧ {𝐵, 𝐴, 𝐶} USGrph 𝐸) ∧ {𝐵, 𝐴} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)
50	40, 44, 49	syl2an 493	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)
51	26, 50	jca 553	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))
52	25, 51	jca 553	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)))
53		preq1 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
54	53	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸))
55		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
56	55	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ ran 𝐸 ↔ {𝐴, 𝑤} ∈ ran 𝐸))
57	56	reubidv 3103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
58	54, 57	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)))
59		preq1 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
60	59	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
61		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
62	61	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ ran 𝐸 ↔ {𝐵, 𝑤} ∈ ran 𝐸))
63	62	reubidv 3103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))
64	60, 63	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸)))
65	58, 64	ralprg 4181	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
66	65	3adant3 1074	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
67	66	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
68	67	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
69	68	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸) ↔ (({𝐴, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸) ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ ran 𝐸))))
70	52, 69	mpbird 246	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
71		diftpsn3 4273	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
72	71	3adant1 1072	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
73		reueq1 3117	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵} → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
74	72, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸))
75	74	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
76	72, 75	raleqbidv 3129	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
77	76	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
78	77	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
79	78	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ ran 𝐸)))
80	70, 79	mpbird 246	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
81	80	3mix3d 1231	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
82		sneq 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
83	82	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
84		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
85	84	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ ran 𝐸 ↔ {𝑣, 𝐴} ∈ ran 𝐸))
86		reueq1 3117	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸))
87	83, 86	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸))
88	85, 87	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → (({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸)))
89	83, 88	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐴 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸)))
90		sneq 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → {ℎ} = {𝐵})
91	90	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
92		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐵 → {𝑣, ℎ} = {𝑣, 𝐵})
93	92	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → ({𝑣, ℎ} ∈ ran 𝐸 ↔ {𝑣, 𝐵} ∈ ran 𝐸))
94		reueq1 3117	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸))
95	91, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸))
96	93, 95	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → (({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸)))
97	91, 96	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐵 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸)))
98		sneq 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → {ℎ} = {𝐶})
99	98	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
100		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐶 → {𝑣, ℎ} = {𝑣, 𝐶})
101	100	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → ({𝑣, ℎ} ∈ ran 𝐸 ↔ {𝑣, 𝐶} ∈ ran 𝐸))
102		reueq1 3117	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
103	99, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))
104	101, 103	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → (({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
105	99, 104	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐶 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸)))
106	89, 97, 105	rextpg 4184	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
107	106	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
108	107	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
109	108	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ ran 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ ran 𝐸))))
110	81, 109	mpbird 246	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))
111	110	ex 449	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
112	4, 111	sylbid 229	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
113	112	expcom 450	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))))
114	113	com23 84	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))))
115	1, 114	mpcom 37	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
116	115	com12 32	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
117		breq1 4586	. . . 4 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 FriendGrph 𝐸 ↔ {𝐴, 𝐵, 𝐶} FriendGrph 𝐸))
118		difeq1 3683	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}))
119		reueq1 3117	. . . . . . . 8 ⊢ ((𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))
120	118, 119	syl 17	. . . . . . 7 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))
121	120	anbi2d 736	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
122	118, 121	raleqbidv 3129	. . . . 5 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
123	122	rexeqbi1dv 3124	. . . 4 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸) ↔ ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))
124	117, 123	imbi12d 333	. . 3 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → ((𝑉 FriendGrph 𝐸 → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))))
125	124	3ad2ant3 1077	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ((𝑉 FriendGrph 𝐸 → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)) ↔ ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸))))
126	116, 125	mpbird 246	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝑉 FriendGrph 𝐸 → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ ran 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∃!wreu 2898 ∖ cdif 3537 {csn 4125 {cpr 4127 {ctp 4129 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: 1to3vfriswmgra 26534

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