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Theorem n4cyclfrgra 26307
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 26281 . . . 4 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
2 4cycl4dv4e 25958 . . . . . . . 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 26280 . . . . . . . . . . . . 13 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
4 simpl 471 . . . . . . . . . . . . . . . . . . . 20 ((𝑐𝑉𝑑𝑉) → 𝑐𝑉)
54adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑐𝑉)
65adantr 479 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑐𝑉)
7 necom 2830 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑐𝑐𝑎)
87biimpi 204 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝑐𝑐𝑎)
983ad2ant2 1075 . . . . . . . . . . . . . . . . . . . 20 ((𝑎𝑏𝑎𝑐𝑎𝑑) → 𝑐𝑎)
109ad2antrl 759 . . . . . . . . . . . . . . . . . . 19 (((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → 𝑐𝑎)
1110adantl 480 . . . . . . . . . . . . . . . . . 18 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑐𝑎)
12 eldifsn 4255 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐𝑉𝑐𝑎))
136, 11, 12sylanbrc 694 . . . . . . . . . . . . . . . . 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 4130 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑎 → {𝑘} = {𝑎})
1514difeq2d 3685 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑎 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑎}))
16 preq2 4208 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 𝑎 → {𝑥, 𝑘} = {𝑥, 𝑎})
1716preq1d 4213 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑎 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑎}, {𝑥, 𝑙}})
1817sseq1d 3590 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑎 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1918reubidv 3098 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑎 → (∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2015, 19raleqbidv 3124 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑎 → (∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2120rspcv 3273 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2221adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑉𝑏𝑉) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2322adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸))
2423adantr 479 . . . . . . . . . . . . . . . . 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 4208 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑐 → {𝑥, 𝑙} = {𝑥, 𝑐})
2625preq2d 4214 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑐 → {{𝑥, 𝑎}, {𝑥, 𝑙}} = {{𝑥, 𝑎}, {𝑥, 𝑐}})
2726sseq1d 3590 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑐 → ({{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
2827reubidv 3098 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑐 → (∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
2928rspcv 3273 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ (𝑉 ∖ {𝑎}) → (∀𝑙 ∈ (𝑉 ∖ {𝑎})∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑙}} ⊆ ran 𝐸 → ∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸))
3013, 24, 29sylsyld 58 . . . . . . . . . . . . . . . 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 4206 . . . . . . . . . . . . . . . . . . . 20 {𝑥, 𝑎} = {𝑎, 𝑥}
3231preq1i 4210 . . . . . . . . . . . . . . . . . . 19 {{𝑥, 𝑎}, {𝑥, 𝑐}} = {{𝑎, 𝑥}, {𝑥, 𝑐}}
3332sseq1i 3587 . . . . . . . . . . . . . . . . . 18 ({{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸 ↔ {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
3433reubii 3100 . . . . . . . . . . . . . . . . 17 (∃!𝑥𝑉 {{𝑥, 𝑎}, {𝑥, 𝑐}} ⊆ ran 𝐸 ↔ ∃!𝑥𝑉 {{𝑎, 𝑥}, {𝑥, 𝑐}} ⊆ ran 𝐸)
35 simpl 471 . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
4039adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑏𝑉)
4140adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑏𝑉)
42 simpr 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑐𝑉𝑑𝑉) → 𝑑𝑉)
4342adantl 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) → 𝑑𝑉)
4443adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑑𝑉)
45 simprr2 1102 . . . . . . . . . . . . . . . . . . . . 21 (((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))) → 𝑏𝑑)
4645adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑎𝑉𝑏𝑉) ∧ (𝑐𝑉𝑑𝑉)) ∧ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑)))) → 𝑏𝑑)
47 4cycl2vnunb 26306 . . . . . . . . . . . . . . . . . . . 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 1329 . . . . . . . . . . . . . . . . . . 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 116 . . . . . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . . . . 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 52 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 448 . . . . . . . . . 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 3015 . . . . . . . . 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 3013 . . . . . . . 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 1255 . . . . . 6 (𝑉 USGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → (𝑉 FriendGrph 𝐸 → (#‘𝐹) ≠ 4))))
6261com34 88 . . . . 5 (𝑉 USGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 FriendGrph 𝐸 → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4))))
6362com23 83 . . . 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 443 . 2 ((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → ((#‘𝐹) = 4 → (#‘𝐹) ≠ 4))
66 df-ne 2777 . . 3 ((#‘𝐹) ≠ 4 ↔ ¬ (#‘𝐹) = 4)
6766biimpri 216 . 2 (¬ (#‘𝐹) = 4 → (#‘𝐹) ≠ 4)
6865, 67pm2.61d1 169 1 ((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 4)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  ∃!wreu 2893  cdif 3532  wss 3535  {csn 4120  {cpr 4122   class class class wbr 4573  ran crn 5025  cfv 5786  (class class class)co 6523  4c4 10915  #chash 12930   USGrph cusg 25621   Cycles ccycl 25797   FriendGrph cfrgra 26277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-usgra 25624  df-wlk 25798  df-trail 25799  df-pth 25800  df-cycl 25803  df-frgra 26278
This theorem is referenced by: (None)
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