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Theorem 3cyclfrgrarn1 26332
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 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → ∃𝑏𝑉𝑐𝑉 ({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.
StepHypRef Expression
1 2pthfrgrarn2 26330 . . 3 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)))
2 necom 2834 . . . . . . . . . . . 12 (𝐴𝐶𝐶𝐴)
3 eldifsn 4259 . . . . . . . . . . . . 13 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
43simplbi2com 654 . . . . . . . . . . . 12 (𝐶𝐴 → (𝐶𝑉𝐶 ∈ (𝑉 ∖ {𝐴})))
52, 4sylbi 205 . . . . . . . . . . 11 (𝐴𝐶 → (𝐶𝑉𝐶 ∈ (𝑉 ∖ {𝐴})))
65com12 32 . . . . . . . . . 10 (𝐶𝑉 → (𝐴𝐶𝐶 ∈ (𝑉 ∖ {𝐴})))
76adantl 480 . . . . . . . . 9 ((𝐴𝑉𝐶𝑉) → (𝐴𝐶𝐶 ∈ (𝑉 ∖ {𝐴})))
87imp 443 . . . . . . . 8 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
9 sneq 4134 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109difeq2d 3689 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
11 preq1 4211 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐴 → {𝑎, 𝑥} = {𝐴, 𝑥})
1211eleq1d 2671 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → ({𝑎, 𝑥} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
1312anbi1d 736 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸)))
14 neeq1 2843 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → (𝑎𝑥𝐴𝑥))
1514anbi1d 736 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → ((𝑎𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
1613, 15anbi12d 742 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
1716rexbidv 3033 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) ↔ ∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
1810, 17raleqbidv 3128 . . . . . . . . . . 11 (𝑎 = 𝐴 → (∀𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) ↔ ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
1918rspcv 3277 . . . . . . . . . 10 (𝐴𝑉 → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
2019adantr 479 . . . . . . . . 9 ((𝐴𝑉𝐶𝑉) → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
2120adantr 479 . . . . . . . 8 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → ∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
22 preq2 4212 . . . . . . . . . . . . 13 (𝑧 = 𝐶 → {𝑥, 𝑧} = {𝑥, 𝐶})
2322eleq1d 2671 . . . . . . . . . . . 12 (𝑧 = 𝐶 → ({𝑥, 𝑧} ∈ ran 𝐸 ↔ {𝑥, 𝐶} ∈ ran 𝐸))
2423anbi2d 735 . . . . . . . . . . 11 (𝑧 = 𝐶 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ↔ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸)))
25 neeq2 2844 . . . . . . . . . . . 12 (𝑧 = 𝐶 → (𝑥𝑧𝑥𝐶))
2625anbi2d 735 . . . . . . . . . . 11 (𝑧 = 𝐶 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐶)))
2724, 26anbi12d 742 . . . . . . . . . 10 (𝑧 = 𝐶 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
2827rexbidv 3033 . . . . . . . . 9 (𝑧 = 𝐶 → (∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧)) ↔ ∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
2928rspcv 3277 . . . . . . . 8 (𝐶 ∈ (𝑉 ∖ {𝐴}) → (∀𝑧 ∈ (𝑉 ∖ {𝐴})∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧)) → ∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
308, 21, 29sylsyld 58 . . . . . . 7 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → ∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
31 2pthfrgrarn 26329 . . . . . . . . . 10 (𝑉 FriendGrph 𝐸 → ∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸))
32 necom 2834 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑥𝑥𝐴)
33 eldifsn 4259 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
3433simplbi2com 654 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴 → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3532, 34sylbi 205 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑥 → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3635adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑥𝐴𝑉) → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3736imp 443 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑥𝐴𝑉) ∧ 𝑥𝑉) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
38 sneq 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 = 𝐴 → {𝑢} = {𝐴})
3938difeq2d 3689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 = 𝐴 → (𝑉 ∖ {𝑢}) = (𝑉 ∖ {𝐴}))
40 preq1 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = 𝐴 → {𝑢, 𝑦} = {𝐴, 𝑦})
4140eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢 = 𝐴 → ({𝑢, 𝑦} ∈ ran 𝐸 ↔ {𝐴, 𝑦} ∈ ran 𝐸))
4241anbi1d 736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 = 𝐴 → (({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4342rexbidv 3033 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 = 𝐴 → (∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4439, 43raleqbidv 3128 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4544rspcv 3277 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4645adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑥𝐴𝑉) → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4746adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑥𝐴𝑉) ∧ 𝑥𝑉) → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
48 preq2 4212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = 𝑥 → {𝑦, 𝑣} = {𝑦, 𝑥})
4948eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑥 → ({𝑦, 𝑣} ∈ ran 𝐸 ↔ {𝑦, 𝑥} ∈ ran 𝐸))
5049anbi2d 735 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
5150rexbidv 3033 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑥 → (∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
5251rspcv 3277 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ (𝑉 ∖ {𝐴}) → (∀𝑣 ∈ (𝑉 ∖ {𝐴})∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
5337, 47, 52sylsyld 58 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑥𝐴𝑉) ∧ 𝑥𝑉) → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
54 prcom 4210 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴, 𝑦} = {𝑦, 𝐴}
5554eleq1i 2678 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝐴, 𝑦} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸)
56 prcom 4210 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑦, 𝑥} = {𝑥, 𝑦}
5756eleq1i 2678 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑦, 𝑥} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸)
5855, 57anbi12ci 729 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) ↔ ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸))
59 preq2 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 = 𝑥 → {𝐴, 𝑏} = {𝐴, 𝑥})
6059eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑥 → ({𝐴, 𝑏} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
61 preq1 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 = 𝑥 → {𝑏, 𝑐} = {𝑥, 𝑐})
6261eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑥 → ({𝑏, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑐} ∈ ran 𝐸))
63 biidd 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑥 → ({𝑐, 𝐴} ∈ ran 𝐸 ↔ {𝑐, 𝐴} ∈ ran 𝐸))
6460, 62, 633anbi123d 1390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑦 → ({𝐴, 𝑥} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
66 preq2 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑐 = 𝑦 → {𝑥, 𝑐} = {𝑥, 𝑦})
6766eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸))
68 preq1 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑐 = 𝑦 → {𝑐, 𝐴} = {𝑦, 𝐴})
6968eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑦 → ({𝑐, 𝐴} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸))
7065, 67, 693anbi123d 1390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
7164, 70rspc2ev 3294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥𝑉𝑦𝑉 ∧ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
72713expa 1256 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥𝑉𝑦𝑉) ∧ ({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
7372expcom 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸) → ((𝑥𝑉𝑦𝑉) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))
74733expib 1259 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝐴, 𝑥} ∈ ran 𝐸 → (({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝐴} ∈ ran 𝐸) → ((𝑥𝑉𝑦𝑉) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
7558, 74syl5bi 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({𝐴, 𝑥} ∈ ran 𝐸 → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → ((𝑥𝑉𝑦𝑉) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
7675adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → ((𝑥𝑉𝑦𝑉) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
7776com13 85 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑉𝑦𝑉) → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
7877rexlimdva 3012 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑉 → (∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
7978com13 85 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
8053, 79syl9 74 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑥𝐴𝑉) ∧ 𝑥𝑉) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
8180exp31 627 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑥 → (𝐴𝑉 → (𝑥𝑉 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
8281com24 92 . . . . . . . . . . . . . . . . . 18 (𝐴𝑥 → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (𝑥𝑉 → (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
8382adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐴𝑥𝑥𝐶) → (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) → (𝑥𝑉 → (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))))
8483impcom 444 . . . . . . . . . . . . . . . 16 ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → (𝑥𝑉 → (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))))
8584com15 98 . . . . . . . . . . . . . . 15 (𝑥𝑉 → (𝑥𝑉 → (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))))
8685pm2.43i 49 . . . . . . . . . . . . . 14 (𝑥𝑉 → (𝐴𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
8786com12 32 . . . . . . . . . . . . 13 (𝐴𝑉 → (𝑥𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
8887adantr 479 . . . . . . . . . . . 12 ((𝐴𝑉𝐶𝑉) → (𝑥𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
8988adantr 479 . . . . . . . . . . 11 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑥𝑉 → (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9089com4t 90 . . . . . . . . . 10 (∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9131, 90syl 17 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑥𝑉 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9291com14 93 . . . . . . . 8 (𝑥𝑉 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9392rexlimiv 3008 . . . . . . 7 (∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶)) → (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
9430, 93syl6 34 . . . . . 6 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (𝑉 FriendGrph 𝐸 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9594pm2.43a 51 . . . . 5 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → (𝑉 FriendGrph 𝐸 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
9695ex 448 . . . 4 ((𝐴𝑉𝐶𝑉) → (𝐴𝐶 → (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → (𝑉 FriendGrph 𝐸 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
9796com4t 90 . . 3 (∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) → (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉) → (𝐴𝐶 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸)))))
981, 97mpcom 37 . 2 (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉) → (𝐴𝐶 → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))))
99983imp 1248 1 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  {csn 4124  {cpr 4126   class class class wbr 4577  ran crn 5028   FriendGrph cfrgra 26308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-n0 11142  df-z 11213  df-uz 11522  df-fz 12155  df-hash 12937  df-usgra 25655  df-frgra 26309
This theorem is referenced by:  3cyclfrgrarn  26333
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