Theorem 3cyclfrgrarn1 26539

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.)

Assertion

Ref	Expression
3cyclfrgrarn1	⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))

Distinct variable groups:   𝑏,𝑐,𝐴   𝐸,𝑐,𝑏   𝑉,𝑐,𝑏

Allowed substitution hints:   𝐶(𝑏,𝑐)

Proof of Theorem 3cyclfrgrarn1

Dummy variables 𝑎 𝑥 𝑧 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2pthfrgrarn2 26537	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)))
2		necom 2835	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
3		eldifsn 4260	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
4	3	simplbi2com 655	. . . . . . . . . . . 12 ⊢ (𝐶 ≠ 𝐴 → (𝐶 ∈ 𝑉 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
5	2, 4	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐴 ≠ 𝐶 → (𝐶 ∈ 𝑉 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
6	5	com12 32	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑉 → (𝐴 ≠ 𝐶 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
7	6	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → 𝐶 ∈ (𝑉 ∖ {𝐴})))
8	7	imp 444	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
9		sneq 4135	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
10	9	difeq2d 3690	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
11		preq1 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐴 → {𝑎, 𝑥} = {𝐴, 𝑥})
12	11	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → ({𝑎, 𝑥} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
13	12	anbi1d 737	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸)))
14		neeq1 2844	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑥 ↔ 𝐴 ≠ 𝑥))
15	14	anbi1d 737	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐴 → ((𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧) ↔ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)))
16	13, 15	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
17	16	rexbidv 3034	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
18	10, 17	raleqbidv 3129	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
19	18	rspcv 3278	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
21	20	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧))))
22		preq2 4213	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → {𝑥, 𝑧} = {𝑥, 𝐶})
23	22	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → ({𝑥, 𝑧} ∈ ran 𝐸 ↔ {𝑥, 𝐶} ∈ ran 𝐸))
24	23	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸)))
25		neeq2 2845	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑥 ≠ 𝑧 ↔ 𝑥 ≠ 𝐶))
26	25	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → ((𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧) ↔ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)))
27	24, 26	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
28	27	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) ↔ ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
29	28	rspcv 3278	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) → (∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
30	8, 21, 29	sylsyld 59	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → ∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶))))
31		2pthfrgrarn 26536	. . . . . . . . . 10 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸))
32		necom 2835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ≠ 𝑥 ↔ 𝑥 ≠ 𝐴)
33		eldifsn 4260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴))
34	33	simplbi2com 655	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ≠ 𝐴 → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
35	32, 34	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ≠ 𝑥 → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
37	36	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
38		sneq 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 = 𝐴 → {𝑢} = {𝐴})
39	38	difeq2d 3690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑢 = 𝐴 → (𝑉 ∖ {𝑢}) = (𝑉 ∖ {𝐴}))
40		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = 𝐴 → {𝑢, 𝑦} = {𝐴, 𝑦})
41	40	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑢 = 𝐴 → ({𝑢, 𝑦} ∈ ran 𝐸 ↔ {𝐴, 𝑦} ∈ ran 𝐸))
42	41	anbi1d 737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 = 𝐴 → (({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
43	42	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑢 = 𝐴 → (∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
44	39, 43	raleqbidv 3129	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
45	44	rspcv 3278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
48		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = 𝑥 → {𝑦, 𝑣} = {𝑦, 𝑥})
49	48	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = 𝑥 → ({𝑦, 𝑣} ∈ ran 𝐸 ↔ {𝑦, 𝑥} ∈ ran 𝐸))
50	49	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
51	50	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = 𝑥 → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
52	51	rspcv 3278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) → (∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
53	37, 47, 52	sylsyld 59	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
54		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝐴, 𝑦} = {𝑦, 𝐴}
55	54	eleq1i 2679	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝐴, 𝑦} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸)
56		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦, 𝑥} = {𝑥, 𝑦}
57	56	eleq1i 2679	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑦, 𝑥} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸)
58	55, 57	anbi12ci 730	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) ↔ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸))
59		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 = 𝑥 → {𝐴, 𝑏} = {𝐴, 𝑥})
60	59	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑥 → ({𝐴, 𝑏} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
61		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 = 𝑥 → {𝑏, 𝑐} = {𝑥, 𝑐})
62	61	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑥 → ({𝑏, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑐} ∈ ran 𝐸))
63		biidd 251	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑥 → ({𝑐, 𝐴} ∈ ran 𝐸 ↔ {𝑐, 𝐴} ∈ ran 𝐸))
64	60, 62, 63	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑥 → (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))
65		biidd 251	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑦 → ({𝐴, 𝑥} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
66		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑐 = 𝑦 → {𝑥, 𝑐} = {𝑥, 𝑦})
67	66	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸))
68		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑐 = 𝑦 → {𝑐, 𝐴} = {𝑦, 𝐴})
69	68	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑐 = 𝑦 → ({𝑐, 𝐴} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸))
70	65, 67, 69	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 = 𝑦 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸)))
71	64, 70	rspc2ev 3295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
72	71	3expa 1257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
73	72	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))
74	73	3expib 1260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({𝐴, 𝑥} ∈ ran 𝐸 → (({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
75	58, 74	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({𝐴, 𝑥} ∈ ran 𝐸 → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
76	75	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
77	76	com13 86	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
78	77	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑉 → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
79	78	com13 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∃𝑦 ∈ 𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
80	53, 79	syl9 75	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ≠ 𝑥 ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
81	80	exp31 628	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ≠ 𝑥 → (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
82	81	com24 93	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ≠ 𝑥 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
83	82	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
84	83	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))))
85	84	com15 99	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))))
86	85	pm2.43i 50	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
87	86	com12 32	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
88	87	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
89	88	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
90	89	com4t 91	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦 ∈ 𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
91	31, 90	syl 17	. . . . . . . . 9 ⊢ (𝑉 FriendGrph 𝐸 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑥 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
92	91	com14 94	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
93	92	rexlimiv 3009	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴 ≠ 𝑥 ∧ 𝑥 ≠ 𝐶)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
94	30, 93	syl6 34	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
95	94	pm2.43a 52	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝑉 FriendGrph 𝐸 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
96	95	ex 449	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝑉 FriendGrph 𝐸 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
97	96	com4t 91	. . 3 ⊢ (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥 ∈ 𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎 ≠ 𝑥 ∧ 𝑥 ≠ 𝑧)) → (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
98	1, 97	mpcom 37	. 2 ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
99	98	3imp 1249	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   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:  3cyclfrgrarn  26540