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Theorem 3vfriswmgr 30366

Description: Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.)

**Hypotheses**
Ref	Expression
3vfriswmgr.v	⊢ 𝑉 = (Vtx‘𝐺)
3vfriswmgr.e	⊢ 𝐸 = (Edg‘𝐺)

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

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

**Proof of Theorem 3vfriswmgr**
Step	Hyp	Ref	Expression
1		frgrusgr 30349	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2		3vfriswmgr.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
3		3vfriswmgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	frgr3v 30363	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))
5	4	exp4b 430	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (𝑉 = {𝐴, 𝐵, 𝐶} → (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸))))))
6	5	3imp1 1349	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
7		prcom 4677	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
8	7	eleq1i 2828	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
9	8	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ({𝐶, 𝐴} ∈ 𝐸 → {𝐴, 𝐶} ∈ 𝐸)
10	9	3ad2ant3 1136	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐴, 𝐶} ∈ 𝐸)
11	10	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → {𝐴, 𝐶} ∈ 𝐸)
12		simpl11 1250	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐴 ∈ 𝑋)
13		simpl12 1251	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐵 ∈ 𝑌)
14		simp1 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐵)
15	14	3ad2ant2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐴 ≠ 𝐵)
16	15	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐴 ≠ 𝐵)
17	12, 13, 16	3jca 1129	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵))
18		simp3 1139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐴, 𝐵, 𝐶})
19	18	anim1i 616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph))
20	17, 19	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)))
21		simp1 1137	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐴, 𝐵} ∈ 𝐸)
22	2, 3	3vfriswmgrlem 30365	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
23	22	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
24	20, 21, 23	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)
25	11, 24	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
26		simpr2 1197	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → {𝐵, 𝐶} ∈ 𝐸)
27		necom 2986	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
28	27	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
29	28	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐴)
30	29	3ad2ant2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝐵 ≠ 𝐴)
31	30	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐵 ≠ 𝐴)
32	13, 12, 31	3jca 1129	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴))
33		tpcoma 4695	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
34	18, 33	eqtrdi 2788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐵, 𝐴, 𝐶})
35	34	anim1i 616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph))
36	32, 35	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → ((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)))
37		prcom 4677	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
38	37	eleq1i 2828	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
39	38	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → {𝐵, 𝐴} ∈ 𝐸)
40	39	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐵, 𝐴} ∈ 𝐸)
41	2, 3	3vfriswmgrlem 30365	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐵, 𝐴} ∈ 𝐸 → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸))
42	41	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐵, 𝐴} ∈ 𝐸) → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸)
43		reueq1 3375	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} = {𝐵, 𝐴} → (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸))
44	37, 43	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸)
45	42, 44	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ {𝐵, 𝐴} ∈ 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)
46	36, 40, 45	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)
47	26, 46	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
48	25, 47	jca 511	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)))
49		preq1 4678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
50	49	eleq1d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
51		preq1 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
52	51	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑤} ∈ 𝐸))
53	52	reubidv 3359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
54	50, 53	anbi12d 633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)))
55		preq1 4678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
56	55	eleq1d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ 𝐸))
57		preq1 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
58	57	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐵, 𝑤} ∈ 𝐸))
59	58	reubidv 3359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
60	56, 59	anbi12d 633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸)))
61	54, 60	ralprg 4641	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
62	61	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
63	62	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
64	63	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
65	64	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ (({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸) ∧ ({𝐵, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))))
66	48, 65	mpbird 257	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
67		diftpsn3 4746	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
68	67	3adant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
69		reueq1 3375	. . . . . . . . . . . . . . . . 17 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵} → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
70	68, 69	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸))
71	70	anbi2d 631	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
72	68, 71	raleqbidv 3312	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
73	72	3ad2ant2 1135	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
74	73	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
75	74	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ {𝐴, 𝐵} ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸)))
76	66, 75	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
77	76	3mix3d 1340	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
78		sneq 4578	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
79	78	difeq2d 4067	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
80		preq2 4679	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
81	80	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
82		reueq1 3375	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
83	79, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
84	81, 83	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
85	79, 84	raleqbidv 3312	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐴 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
86		sneq 4578	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → {ℎ} = {𝐵})
87	86	difeq2d 4067	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
88		preq2 4679	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐵 → {𝑣, ℎ} = {𝑣, 𝐵})
89	88	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐵} ∈ 𝐸))
90		reueq1 3375	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸))
91	87, 90	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸))
92	89, 91	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸)))
93	87, 92	raleqbidv 3312	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐵 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸)))
94		sneq 4578	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → {ℎ} = {𝐶})
95	94	difeq2d 4067	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
96		preq2 4679	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐶 → {𝑣, ℎ} = {𝑣, 𝐶})
97	96	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐶} ∈ 𝐸))
98		reueq1 3375	. . . . . . . . . . . . . . . 16 ⊢ (({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
99	95, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → (∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))
100	97, 99	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
101	95, 100	raleqbidv 3312	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝐶 → (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸)))
102	85, 93, 101	rextpg 4644	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
103	102	3ad2ant1 1134	. . . . . . . . . . 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} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
104	103	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
105	104	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → (∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴})({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵})({𝑣, 𝐵} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐵}){𝑣, 𝑤} ∈ 𝐸) ∨ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶})({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝐶}){𝑣, 𝑤} ∈ 𝐸))))
106	77, 105	mpbird 257	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
107	106	ex 412	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
108	6, 107	sylbid 240	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
109	108	expcom 413	. . . . 5 ⊢ (𝐺 ∈ USGraph → (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))))
110	109	com23 86	. . . 4 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (((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} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))))
111	1, 110	mpcom 38	. . 3 ⊢ (𝐺 ∈ FriendGraph → (((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} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
112	111	com12 32	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
113		difeq1 4060	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}))
114		reueq1 3375	. . . . . . . 8 ⊢ ((𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
115	113, 114	syl 17	. . . . . . 7 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
116	115	anbi2d 631	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
117	113, 116	raleqbidv 3312	. . . . 5 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
118	117	rexeqbi1dv 3307	. . . 4 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
119	118	imbi2d 340	. . 3 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → ((𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)) ↔ (𝐺 ∈ FriendGraph → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))))
120	119	3ad2ant3 1136	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → ((𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)) ↔ (𝐺 ∈ FriendGraph → ∃ℎ ∈ {𝐴, 𝐵, 𝐶}∀𝑣 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴, 𝐵, 𝐶} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))))
121	112, 120	mpbird 257	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∃!wreu 3341 ∖ cdif 3887 {csn 4568 {cpr 4570 {ctp 4572 ‘cfv 6493 Vtxcvtx 29082 Edgcedg 29133 USGraphcusgr 29235 FriendGraph cfrgr 30346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-hash 14287 df-edg 29134 df-umgr 29169 df-usgr 29237 df-frgr 30347

This theorem is referenced by: 1to3vfriswmgr 30368

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