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Theorem n4cyclfrgra 26545
 Description: There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
Assertion
Ref Expression
n4cyclfrgra ((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 4)

Proof of Theorem n4cyclfrgra
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26519 . . . 4 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
2 4cycl4dv4e 26196 . . . . . . . 8 ((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))))
3 frisusgrapr 26518 . . . . . . . . . . . . 13 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
4 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑐𝑉𝑑𝑉) → 𝑐𝑉)
54adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑐𝑉)
65adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑐𝑉)
7 necom 2835 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑐𝑐𝑎)
87biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝑐𝑐𝑎)
983ad2ant2 1076 . . . . . . . . . . . . . . . . . . . 20 ((𝑎𝑏𝑎𝑐𝑎𝑑) → 𝑐𝑎)
109ad2antrl 760 . . . . . . . . . . . . . . . . . . 19 (((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → 𝑐𝑎)
1110adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑐𝑎)
12 eldifsn 4260 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐𝑉𝑐𝑎))
136, 11, 12sylanbrc 695 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑐 ∈ (𝑉 ∖ {𝑎}))
14 sneq 4135 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑎 → {𝑘} = {𝑎})
1514difeq2d 3690 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑎 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑎}))
16 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑎 → {𝑥, 𝑘} = {𝑥, 𝑎})
1716preq1d 4218 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑎 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑎}, {𝑥, 𝑙}})
1817sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑎 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1918reubidv 3103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑎 → (∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2015, 19raleqbidv 3129 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑎 → (∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2120rspcv 3278 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2221adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑉𝑏𝑉) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2322adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2423adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
25 preq2 4213 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑐 → {𝑥, 𝑙} = {𝑥, 𝑐})
2625preq2d 4219 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑐 → {{𝑥, 𝑎}, {𝑥, 𝑙}} = {{𝑥, 𝑎}, {𝑥, 𝑐}})
2726sseq1d 3595 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑐 → ({{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
2827reubidv 3103 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑐 → (∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
2928rspcv 3278 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ (𝑉 ∖ {𝑎}) → (∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
3013, 24, 29sylsyld 59 . . . . . . . . . . . . . . . 16 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
31 prcom 4211 . . . . . . . . . . . . . . . . . . . 20 {𝑥, 𝑎} = {𝑎, 𝑥}
3231preq1i 4215 . . . . . . . . . . . . . . . . . . 19 {{𝑥, 𝑎}, {𝑥, 𝑐}} = {{𝑎, 𝑥}, {𝑥, 𝑐}}
3332sseq1i 3592 . . . . . . . . . . . . . . . . . 18 ({{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸 ↔ {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
3433reubii 3105 . . . . . . . . . . . . . . . . 17 (∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
35 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
3635ad2antrl 760 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
37 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) → ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))
3837ad2antrl 760 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))
39 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
4039adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑏𝑉)
4140adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑏𝑉)
42 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐𝑉𝑑𝑉) → 𝑑𝑉)
4342adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑑𝑉)
4443adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑑𝑉)
45 simprr2 1103 . . . . . . . . . . . . . . . . . . . . 21 (((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → 𝑏𝑑)
4645adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑏𝑑)
47 4cycl2vnunb 26544 . . . . . . . . . . . . . . . . . . . 20 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸) ∧ (𝑏𝑉𝑑𝑉𝑏𝑑)) → ¬ ∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
4836, 38, 41, 44, 46, 47syl113anc 1330 . . . . . . . . . . . . . . . . . . 19 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → ¬ ∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
4948pm2.21d 117 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸 → (#‘𝐹) ≠ 4))
5049com12 32 . . . . . . . . . . . . . . . . 17 (∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸 → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4))
5134, 50sylbi 206 . . . . . . . . . . . . . . . 16 (∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸 → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4))
5230, 51syl6 34 . . . . . . . . . . . . . . 15 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4)))
5352pm2.43b 53 . . . . . . . . . . . . . 14 (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4))
5453adantl 481 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸) → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4))
553, 54syl 17 . . . . . . . . . . . 12 (𝑉 FriendGrph 𝐸 → ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (#‘𝐹) ≠ 4))
5655com12 32 . . . . . . . . . . 11 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4))
5756ex 449 . . . . . . . . . 10 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → (((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4)))
5857rexlimdvva 3020 . . . . . . . . 9 ((𝑎𝑉𝑏𝑉) → (∃𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4)))
5958rexlimivv 3018 . . . . . . . 8 (∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4))
602, 59syl 17 . . . . . . 7 ((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4))
61603exp 1256 . . . . . 6 (𝑉 USGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4))))
6261com34 89 . . . . 5 (𝑉 USGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 FriendGrph 𝐸 → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4))))
6362com23 84 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 FriendGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4))))
641, 63mpcom 37 . . 3 (𝑉 FriendGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4)))
6564imp 444 . 2 ((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4))
66 df-ne 2782 . . 3 ((#‘𝐹) ≠ 4 ↔ ¬ (#‘𝐹) = 4)
6766biimpri 217 . 2 (¬ (#‘𝐹) = 4 → (#‘𝐹) ≠ 4)
6865, 67pm2.61d1 170 1 ((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 4)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  4c4 10949  #chash 12979   USGrph cusg 25859   Cycles ccycl 26035   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-map 7746  df-pm 7747  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-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-cycl 26041  df-frgra 26516 This theorem is referenced by: (None)
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