Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3cyclfrgrarn1 Structured version   Visualization version   GIF version

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 ∧ (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 26537 . . 3 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑧 ∈ (𝑉 ∖ {𝑎})∃𝑥𝑉 (({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)))
2 necom 2835 . . . . . . . . . . . 12 (𝐴𝐶𝐶𝐴)
3 eldifsn 4260 . . . . . . . . . . . . 13 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
43simplbi2com 655 . . . . . . . . . . . 12 (𝐶𝐴 → (𝐶𝑉𝐶 ∈ (𝑉 ∖ {𝐴})))
52, 4sylbi 206 . . . . . . . . . . 11 (𝐴𝐶 → (𝐶𝑉𝐶 ∈ (𝑉 ∖ {𝐴})))
65com12 32 . . . . . . . . . 10 (𝐶𝑉 → (𝐴𝐶𝐶 ∈ (𝑉 ∖ {𝐴})))
76adantl 481 . . . . . . . . 9 ((𝐴𝑉𝐶𝑉) → (𝐴𝐶𝐶 ∈ (𝑉 ∖ {𝐴})))
87imp 444 . . . . . . . 8 (((𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
9 sneq 4135 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109difeq2d 3690 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
11 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐴 → {𝑎, 𝑥} = {𝐴, 𝑥})
1211eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → ({𝑎, 𝑥} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
1312anbi1d 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 → (𝑎𝑥𝐴𝑥))
1514anbi1d 737 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → ((𝑎𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝑧)))
1613, 15anbi12d 743 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((({𝑎, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝑎𝑥𝑥𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧))))
1716rexbidv 3034 . . . . . . . . . . . 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 3129 . . . . . . . . . . 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 3278 . . . . . . . . . 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 480 . . . . . . . . 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 480 . . . . . . . 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 4213 . . . . . . . . . . . . 13 (𝑧 = 𝐶 → {𝑥, 𝑧} = {𝑥, 𝐶})
2322eleq1d 2672 . . . . . . . . . . . 12 (𝑧 = 𝐶 → ({𝑥, 𝑧} ∈ ran 𝐸 ↔ {𝑥, 𝐶} ∈ ran 𝐸))
2423anbi2d 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 → (𝑥𝑧𝑥𝐶))
2625anbi2d 736 . . . . . . . . . . 11 (𝑧 = 𝐶 → ((𝐴𝑥𝑥𝑧) ↔ (𝐴𝑥𝑥𝐶)))
2724, 26anbi12d 743 . . . . . . . . . 10 (𝑧 = 𝐶 → ((({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧)) ↔ (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
2827rexbidv 3034 . . . . . . . . 9 (𝑧 = 𝐶 → (∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝑧} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝑧)) ↔ ∃𝑥𝑉 (({𝐴, 𝑥} ∈ ran 𝐸 ∧ {𝑥, 𝐶} ∈ ran 𝐸) ∧ (𝐴𝑥𝑥𝐶))))
2928rspcv 3278 . . . . . . . 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 59 . . . . . . 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 26536 . . . . . . . . . 10 (𝑉 FriendGrph 𝐸 → ∀𝑢𝑉𝑣 ∈ (𝑉 ∖ {𝑢})∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸))
32 necom 2835 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑥𝑥𝐴)
33 eldifsn 4260 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
3433simplbi2com 655 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴 → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3532, 34sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑥 → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3635adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴𝑥𝐴𝑉) → (𝑥𝑉𝑥 ∈ (𝑉 ∖ {𝐴})))
3736imp 444 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑥𝐴𝑉) ∧ 𝑥𝑉) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
38 sneq 4135 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 = 𝐴 → {𝑢} = {𝐴})
3938difeq2d 3690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 = 𝐴 → (𝑉 ∖ {𝑢}) = (𝑉 ∖ {𝐴}))
40 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑢 = 𝐴 → {𝑢, 𝑦} = {𝐴, 𝑦})
4140eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑢 = 𝐴 → ({𝑢, 𝑦} ∈ ran 𝐸 ↔ {𝐴, 𝑦} ∈ ran 𝐸))
4241anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑢 = 𝐴 → (({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4342rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑢 = 𝐴 → (∃𝑦𝑉 ({𝑢, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸)))
4439, 43raleqbidv 3129 . . . . . . . . . . . . . . . . . . . . . . . . 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 3278 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 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 4213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = 𝑥 → {𝑦, 𝑣} = {𝑦, 𝑥})
4948eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑥 → ({𝑦, 𝑣} ∈ ran 𝐸 ↔ {𝑦, 𝑥} ∈ ran 𝐸))
5049anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
5150rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑥 → (∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑣} ∈ ran 𝐸) ↔ ∃𝑦𝑉 ({𝐴, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑥} ∈ ran 𝐸)))
5251rspcv 3278 . . . . . . . . . . . . . . . . . . . . . 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 59 . . . . . . . . . . . . . . . . . . . . 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 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴, 𝑦} = {𝑦, 𝐴}
5554eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝐴, 𝑦} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸)
56 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑦, 𝑥} = {𝑥, 𝑦}
5756eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({𝑦, 𝑥} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸)
5855, 57anbi12ci 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})
6059eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑥 → ({𝐴, 𝑏} ∈ ran 𝐸 ↔ {𝐴, 𝑥} ∈ ran 𝐸))
61 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 = 𝑥 → {𝑏, 𝑐} = {𝑥, 𝑐})
6261eleq1d 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))
6460, 62, 633anbi123d 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})
6766eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑦 → ({𝑥, 𝑐} ∈ ran 𝐸 ↔ {𝑥, 𝑦} ∈ ran 𝐸))
68 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑐 = 𝑦 → {𝑐, 𝐴} = {𝑦, 𝐴})
6968eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑐 = 𝑦 → ({𝑐, 𝐴} ∈ ran 𝐸 ↔ {𝑦, 𝐴} ∈ ran 𝐸))
7065, 67, 693anbi123d 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)))
7164, 70rspc2ev 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1260 . . . . . . . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 86 . . . . . . . . . . . . . . . . . . . . . . 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 3013 . . . . . . . . . . . . . . . . . . . . . 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 86 . . . . . . . . . . . . . . . . . . . . 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 75 . . . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . . . 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 93 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . 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 99 . . . . . . . . . . . . . . 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 50 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 91 . . . . . . . . . 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 94 . . . . . . . 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 3009 . . . . . . 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 52 . . . . 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 449 . . . 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 91 . . 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 1249 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 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
 Copyright terms: Public domain W3C validator