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Theorem 3cyclfrgrrn1 30373

Description: Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.)

**Hypotheses**
Ref	Expression
3cyclfrgrrn1.v	⊢ 𝑉 = (Vtx‘𝐺)
3cyclfrgrrn1.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
3cyclfrgrrn1	⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))

Distinct variable groups: 𝐴,𝑏,𝑐 𝐸,𝑏,𝑐 𝑉,𝑏,𝑐 Allowed substitution hints: 𝐶(𝑏,𝑐) 𝐺(𝑏,𝑐)

Proof of Theorem 3cyclfrgrrn1 Dummy variables 𝑎 𝑥 𝑧 𝑦 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3cyclfrgrrn1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		3cyclfrgrrn1.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	2pthfrgrrn2 30371	. . 3 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)))
4		necom 2987	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
5		eldifsn 4719	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
6	5	simplbi2com 503	. . . . . . . . . . . 12 ⊢ (𝐶 ≠ 𝐴 → (𝐶 ∈ 𝑉 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
7	4, 6	sylbi 218	. . . . . . . . . . 11 ⊢ (𝐴 ≠ 𝐶 → (𝐶 ∈ 𝑉 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
8	7	com12 32	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑉 → (𝐴 ≠ 𝐶 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
9	8	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
10	9	imp 407	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
11		sneq 4565	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
12	11	difeq2d 4057	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
13		preq1 4665	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑥} = {𝐴, 𝑥})
14	13	eleq1d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑥} ∈ 𝐸 ↔ {𝐴, 𝑥} ∈ 𝐸))
15	14	anbi1d 637	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ↔ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸)))
16		neeq1 2996	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑥 ↔ 𝐴 ≠ 𝑥))
17	16	anbi1d 637	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧) ↔ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)))
18	15, 17	anbi12d 638	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ((({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
19	18	rexbidv 3163	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
20	12, 19	raleqbidv 3313	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
21	20	rspcv 3556	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
22	21	ad2antrr 732	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
23		preq2 4666	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → {𝑥, 𝑧} = {𝑥, 𝐶})
24	23	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → ({𝑥, 𝑧} ∈ 𝐸 ↔ {𝑥, 𝐶} ∈ 𝐸))
25	24	anbi2d 636	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ↔ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸)))
26		neeq2 2997	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑥 ≠ 𝑧 ↔ 𝑥 ≠ 𝐶))
27	26	anbi2d 636	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → ((𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧) ↔ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)))
28	25, 27	anbi12d 638	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
29	28	rexbidv 3163	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
30	29	rspcv 3556	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) → (∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
31	10, 22, 30	sylsyld 61	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
32	1, 2	2pthfrgrrn 30370	. . . . . . . . . 10 ⊢ (𝐺 ∈ FriendGraph → ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸))
33		necom 2987	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ≠ 𝑥 ↔ 𝑥 ≠ 𝐴)
34		eldifsn 4719	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴))
35	34	simplbi2com 503	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ≠ 𝐴 → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
36	33, 35	sylbi 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ≠ 𝑥 → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
37	36	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
38	37	imp 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
39		sneq 4565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑢 = 𝐴 → {𝑢} = {𝐴})
40	39	difeq2d 4057	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝐴 → (𝑉 ∖ {𝑢}) = (𝑉 ∖ {𝐴}))
41		preq1 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑢 = 𝐴 → {𝑢, 𝑦} = {𝐴, 𝑦})
42	41	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 = 𝐴 → ({𝑢, 𝑦} ∈ 𝐸 ↔ {𝐴, 𝑦} ∈ 𝐸))
43	42	anbi1d 637	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑢 = 𝐴 → (({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) ↔ ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
44	43	rexbidv 3163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝐴 → (∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) ↔ ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
45	40, 44	raleqbidv 3313	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) ↔ ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
46	45	rspcv 3556	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
47	46	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
48	47	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸)))
49		preq2 4666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑥 → {𝑦, 𝑣} = {𝑦, 𝑥})
50	49	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → ({𝑦, 𝑣} ∈ 𝐸 ↔ {𝑦, 𝑥} ∈ 𝐸))
51	50	anbi2d 636	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = 𝑥 → (({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) ↔ ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸)))
52	51	rexbidv 3163	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑥 → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) ↔ ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸)))
53	52	rspcv 3556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) → (∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸)))
54	38, 48, 53	sylsyld 61	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸)))
55		prcom 4664	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {𝐴, 𝑦} = {𝑦, 𝐴}
56	55	eleq1i 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝐴, 𝑦} ∈ 𝐸 ↔ {𝑦, 𝐴} ∈ 𝐸)
57		prcom 4664	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {𝑦, 𝑥} = {𝑥, 𝑦}
58	57	eleq1i 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝑦, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸)
59	56, 58	anbi12ci 635	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) ↔ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸))
60		preq2 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑥 → {𝐴, 𝑏} = {𝐴, 𝑥})
61	60	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑥 → ({𝐴, 𝑏} ∈ 𝐸 ↔ {𝐴, 𝑥} ∈ 𝐸))
62		preq1 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑥 → {𝑏, 𝑐} = {𝑥, 𝑐})
63	62	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑥 → ({𝑏, 𝑐} ∈ 𝐸 ↔ {𝑥, 𝑐} ∈ 𝐸))
64		biidd 263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑥 → ({𝑐, 𝐴} ∈ 𝐸 ↔ {𝑐, 𝐴} ∈ 𝐸))
65	61, 63, 64	3anbi123d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑥 → (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))
66		biidd 263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑦 → ({𝐴, 𝑥} ∈ 𝐸 ↔ {𝐴, 𝑥} ∈ 𝐸))
67		preq2 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑦 → {𝑥, 𝑐} = {𝑥, 𝑦})
68	67	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ 𝐸 ↔ {𝑥, 𝑦} ∈ 𝐸))
69		preq1 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑦 → {𝑐, 𝐴} = {𝑦, 𝐴})
70	69	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑦 → ({𝑐, 𝐴} ∈ 𝐸 ↔ {𝑦, 𝐴} ∈ 𝐸))
71	66, 68, 70	3anbi123d 1444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑦 → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸) ↔ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸)))
72	65, 71	rspc2ev 3573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))
73	72	3expa 1124	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))
74	73	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))
75	74	3expib 1128	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({𝐴, 𝑥} ∈ 𝐸 → (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝐴} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
76	59, 75	biimtrid 243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝐴, 𝑥} ∈ 𝐸 → (({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
77	76	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
78	77	com13 88	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
79	78	rexlimdva 3140	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑉 → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
80	79	com13 88	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑥} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
81	54, 80	syl9 77	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
82	81	exp31 420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ≠ 𝑥 → (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))))
83	82	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≠ 𝑥 → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))))
84	83	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶) → (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))))
85	84	impcom 408	. . . . . . . . . . . . . . 15 ⊢ ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))))
86	85	com15 101	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))))
87	86	pm2.43i 52	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
88	87	com12 32	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
89	88	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
90	89	com4t 93	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑣} ∈ 𝐸) → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
91	32, 90	syl 17	. . . . . . . . 9 ⊢ (𝐺 ∈ FriendGraph → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
92	91	com14 96	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → ((({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
93	92	rexlimiv 3133	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝐶} ∈ 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
94	31, 93	syl6 35	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
95	94	pm2.43a 54	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
96	95	ex 413	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
97	96	com4t 93	. . 3 ⊢ (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)))))
98	3, 97	mpcom 38	. 2 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))))
99	98	3imp 1116	1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 ∖ cdif 3880 {csn 4555 {cpr 4557 ‘cfv 6485 Vtxcvtx 29083 Edgcedg 29134 FriendGraph cfrgr 30346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-hash 14284 df-edg 29135 df-umgr 29170 df-usgr 29238 df-frgr 30347

This theorem is referenced by: 3cyclfrgrrn 30374

W3C validator