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Theorem 3vfriswmgr 30297

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 30280	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2		3vfriswmgr.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
3		3vfriswmgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	frgr3v 30294	. . . . . . . . 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 1348	. . . . . . 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 4732	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
8	7	eleq1i 2832	. . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph) → 𝐴 ∈ 𝑋)
13		simpl12 1250	. . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . 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 30296	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 1196	. . . . . . . . . . . . . 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 2994	. . . . . . . . . . . . . . . . . . . . 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 4750	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
34	18, 33	eqtrdi 2793	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐵, 𝐴, 𝐶})
35	34	anim1i 615	. . . . . . . . . . . . . . . 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 4732	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
38	37	eleq1i 2832	. . . . . . . . . . . . . . . . 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 30296	. . . . . . . . . . . . . . . . 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 3417	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 4733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
50	49	eleq1d 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
51		preq1 4733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
52	51	eleq1d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑤} ∈ 𝐸))
53	52	reubidv 3398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
54	50, 53	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)))
55		preq1 4733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
56	55	eleq1d 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ 𝐸))
57		preq1 4733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
58	57	eleq1d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐵, 𝑤} ∈ 𝐸))
59	58	reubidv 3398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
60	56, 59	anbi12d 632	. . . . . . . . . . . . . . . . 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 4696	. . . . . . . . . . . . . . . 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 4802	. . . . . . . . . . . . . . . 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 3417	. . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 3346	. . . . . . . . . . . . . 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 1339	. . . . . . . . 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 4636	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
79	78	difeq2d 4126	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
80		preq2 4734	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
81	80	eleq1d 2826	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
82		reueq1 3417	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 3346	. . . . . . . . . . . . 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 4636	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → {ℎ} = {𝐵})
87	86	difeq2d 4126	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
88		preq2 4734	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐵 → {𝑣, ℎ} = {𝑣, 𝐵})
89	88	eleq1d 2826	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐵} ∈ 𝐸))
90		reueq1 3417	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 3346	. . . . . . . . . . . . 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 4636	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → {ℎ} = {𝐶})
95	94	difeq2d 4126	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
96		preq2 4734	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐶 → {𝑣, ℎ} = {𝑣, 𝐶})
97	96	eleq1d 2826	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐶} ∈ 𝐸))
98		reueq1 3417	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 3346	. . . . . . . . . . . . 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 4699	. . . . . . . . . . . 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 4119	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}))
114		reueq1 3417	. . . . . . . 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 630	. . . . . 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 3346	. . . . 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 3339	. . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 ∖ cdif 3948 {csn 4626 {cpr 4628 {ctp 4630 ‘cfv 6561 Vtxcvtx 29013 Edgcedg 29064 USGraphcusgr 29166 FriendGraph cfrgr 30277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-edg 29065 df-umgr 29100 df-usgr 29168 df-frgr 30278

This theorem is referenced by: 1to3vfriswmgr 30299

W3C validator