3vfriswmgr - Mathbox for Alexander van der Vekens

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Theorem 3vfriswmgr 41448

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 41432	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
2		3vfriswmgr.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
3		3vfriswmgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	frgr3v 41445	. . . . . . . . 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 630	. . . . . . . 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 1272	. . . . . . 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 4211	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
8	7	eleq1i 2679	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐶, 𝐴} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸)
9	8	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ ({𝐶, 𝐴} ∈ 𝐸 → {𝐴, 𝐶} ∈ 𝐸)
10	9	3ad2ant3 1077	. . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph ) → 𝐴 ∈ 𝑋)
13		simpl12 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ USGraph ) → 𝐵 ∈ 𝑌)
14		simp1 1054	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐵)
15	14	3ad2ant2 1076	. . . . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . 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 1056	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐴, 𝐵, 𝐶})
19	18	anim1i 590	. . . . . . . . . . . . . . . 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 553	. . . . . . . . . . . . . . 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 1054	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐴, 𝐵} ∈ 𝐸)
22	2, 3	3vfriswmgrlem 41447	. . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . 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 553	. . . . . . . . . . . . 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 1061	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
28	27	biimpi 205	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
29	28	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐴)
30	29	3ad2ant2 1076	. . . . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . 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 4229	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
34	18, 33	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → 𝑉 = {𝐵, 𝐴, 𝐶})
35	34	anim1i 590	. . . . . . . . . . . . . . . 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 553	. . . . . . . . . . . . . . 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 4211	. . . . . . . . . . . . . . . . . 18 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
38	37	eleq1i 2679	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐵, 𝐴} ∈ 𝐸)
39	38	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} ∈ 𝐸 → {𝐵, 𝐴} ∈ 𝐸)
40	39	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) → {𝐵, 𝐴} ∈ 𝐸)
41	2, 3	3vfriswmgrlem 41447	. . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph )) ∧ {𝐵, 𝐴} ∈ 𝐸) → ∃!𝑤 ∈ {𝐵, 𝐴} {𝐵, 𝑤} ∈ 𝐸)
43		reueq1 3117	. . . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . 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 553	. . . . . . . . . . . . 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 553	. . . . . . . . . . . 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 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → {𝑣, 𝐶} = {𝐴, 𝐶})
50	49	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐴, 𝐶} ∈ 𝐸))
51		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐴 → {𝑣, 𝑤} = {𝐴, 𝑤})
52	51	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐴 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐴, 𝑤} ∈ 𝐸))
53	52	reubidv 3103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸))
54	50, 53	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐴 → (({𝑣, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸) ↔ ({𝐴, 𝐶} ∈ 𝐸 ∧ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)))
55		preq1 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → {𝑣, 𝐶} = {𝐵, 𝐶})
56	55	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ 𝐸))
57		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝐵 → {𝑣, 𝑤} = {𝐵, 𝑤})
58	57	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝐵 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝐵, 𝑤} ∈ 𝐸))
59	58	reubidv 3103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐵 → (∃!𝑤 ∈ {𝐴, 𝐵} {𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ {𝐴, 𝐵} {𝐵, 𝑤} ∈ 𝐸))
60	56, 59	anbi12d 743	. . . . . . . . . . . . . . . . 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 4181	. . . . . . . . . . . . . . . 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 1074	. . . . . . . . . . . . . . 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 1075	. . . . . . . . . . . . . 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 246	. . . . . . . . . . 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 4273	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
68	67	3adant1 1072	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
69		reueq1 3117	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . . 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 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} ∈ 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 246	. . . . . . . . . 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 1231	. . . . . . . . 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 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
79	78	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
80		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
81	80	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
82		reueq1 3117	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → {ℎ} = {𝐵})
87	86	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
88		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐵 → {𝑣, ℎ} = {𝑣, 𝐵})
89	88	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐵 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐵} ∈ 𝐸))
90		reueq1 3117	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → {ℎ} = {𝐶})
95	94	difeq2d 3690	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
96		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = 𝐶 → {𝑣, ℎ} = {𝑣, 𝐶})
97	96	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝐶 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐶} ∈ 𝐸))
98		reueq1 3117	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . . 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 4184	. . . . . . . . . . . 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 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} ∈ 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 246	. . . . . . . 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 449	. . . . . . 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 229	. . . . . 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 450	. . . . 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 84	. . . 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 37	. . 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 3683	. . . . . 6 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑉 ∖ {ℎ}) = ({𝐴, 𝐵, 𝐶} ∖ {ℎ}))
114		reueq1 3117	. . . . . . . 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 736	. . . . . 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 3129	. . . . 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 3124	. . . 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 329	. . 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 1077	. 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 246	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 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 ‘cfv 5804 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 FriendGraph cfrgr 41428

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-umgr 25750 df-edga 25793 df-usgr 40381 df-frgr 41429

This theorem is referenced by: 1to3vfriswmgr 41450

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