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Theorem usgrcyclnl2 26169
 Description: In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Assertion
Ref Expression
usgrcyclnl2 ((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)

Proof of Theorem usgrcyclnl2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 cyclispth 26157 . . . 4 (𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Paths 𝐸)𝑃)
2 pthistrl 26102 . . . . 5 (𝐹(𝑉 Paths 𝐸)𝑃𝐹(𝑉 Trails 𝐸)𝑃)
3 cycliswlk 26160 . . . . . 6 (𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)
4 wlkbprop 26051 . . . . . 6 (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
5 istrl2 26068 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
6 pm3.2 462 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
75, 6sylbid 229 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
873adant1 1072 . . . . . 6 (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
93, 4, 83syl 18 . . . . 5 (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝐹(𝑉 Trails 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
102, 9syl5com 31 . . . 4 (𝐹(𝑉 Paths 𝐸)𝑃 → (𝐹(𝑉 Cycles 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
111, 10mpcom 37 . . 3 (𝐹(𝑉 Cycles 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
12 iscycl 26153 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
1312adantr 480 . . . 4 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
14 oveq2 6557 . . . . . . . . . . . . . 14 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
15 f1eq2 6010 . . . . . . . . . . . . . 14 ((0..^(#‘𝐹)) = (0..^2) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝐹:(0..^2)–1-1→dom 𝐸))
1614, 15syl 17 . . . . . . . . . . . . 13 ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝐹:(0..^2)–1-1→dom 𝐸))
1714raleqdv 3121 . . . . . . . . . . . . 13 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
1816, 17anbi12d 743 . . . . . . . . . . . 12 ((#‘𝐹) = 2 → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
19 fveq2 6103 . . . . . . . . . . . . 13 ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
2019eqeq2d 2620 . . . . . . . . . . . 12 ((#‘𝐹) = 2 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘2)))
2118, 20anbi12d 743 . . . . . . . . . . 11 ((#‘𝐹) = 2 → (((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ↔ ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2))))
2221anbi1d 737 . . . . . . . . . 10 ((#‘𝐹) = 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) ↔ (((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2)) ∧ 𝑉 USGrph 𝐸)))
23 fzo0to2pr 12420 . . . . . . . . . . . . . 14 (0..^2) = {0, 1}
2423raleqi 3119 . . . . . . . . . . . . 13 (∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
25 2wlklem 26094 . . . . . . . . . . . . . 14 (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
26 prcom 4211 . . . . . . . . . . . . . . . . . . 19 {(𝑃‘1), (𝑃‘2)} = {(𝑃‘2), (𝑃‘1)}
2726eqeq2i 2622 . . . . . . . . . . . . . . . . . 18 ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)})
28 preq1 4212 . . . . . . . . . . . . . . . . . . . 20 ((𝑃‘2) = (𝑃‘0) → {(𝑃‘2), (𝑃‘1)} = {(𝑃‘0), (𝑃‘1)})
2928eqcoms 2618 . . . . . . . . . . . . . . . . . . 19 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘2), (𝑃‘1)} = {(𝑃‘0), (𝑃‘1)})
3029eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 ((𝑃‘0) = (𝑃‘2) → ((𝐸‘(𝐹‘1)) = {(𝑃‘2), (𝑃‘1)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
3127, 30syl5bb 271 . . . . . . . . . . . . . . . . 17 ((𝑃‘0) = (𝑃‘2) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
3231anbi2d 736 . . . . . . . . . . . . . . . 16 ((𝑃‘0) = (𝑃‘2) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
33 eqtr3 2631 . . . . . . . . . . . . . . . . 17 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)))
34 usgraf1 25889 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
35 f1f 6014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0..^2)–1-1→dom 𝐸𝐹:(0..^2)⟶dom 𝐸)
36 2nn 11062 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 ∈ ℕ
37 lbfzo0 12375 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
3836, 37mpbir 220 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ (0..^2)
39 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝐹‘0) ∈ dom 𝐸)
4038, 39mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘0) ∈ dom 𝐸)
41 1nn0 11185 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ ℕ0
42 1lt2 11071 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 < 2
43 elfzo0 12376 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
4441, 36, 42, 43mpbir3an 1237 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (0..^2)
45 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝐹‘1) ∈ dom 𝐸)
4644, 45mpan2 703 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:(0..^2)⟶dom 𝐸 → (𝐹‘1) ∈ dom 𝐸)
4740, 46jca 553 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0..^2)⟶dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
4835, 47syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸))
49 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ((𝐹‘0) ∈ dom 𝐸 ∧ (𝐹‘1) ∈ dom 𝐸)) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) ↔ (𝐹‘0) = (𝐹‘1)))
5048, 49sylan2 490 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸:dom 𝐸1-1→ran 𝐸𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) ↔ (𝐹‘0) = (𝐹‘1)))
51 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))) → ((𝐹‘0) = (𝐹‘1) ↔ 0 = 1))
5238, 44, 51mpanr12 717 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) ↔ 0 = 1))
53 ax-1ne0 9884 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ≠ 0
54 necom 2835 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ≠ 0 ↔ 0 ≠ 1)
55 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ≠ 1 ↔ ¬ 0 = 1)
56 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (¬ 0 = 1 → (0 = 1 → (#‘𝐹) ≠ 2))
5755, 56sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ≠ 1 → (0 = 1 → (#‘𝐹) ≠ 2))
5854, 57sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ≠ 0 → (0 = 1 → (#‘𝐹) ≠ 2))
5953, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (0 = 1 → (#‘𝐹) ≠ 2)
6052, 59syl6bi 242 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐹‘0) = (𝐹‘1) → (#‘𝐹) ≠ 2))
6160adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸:dom 𝐸1-1→ran 𝐸𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐹‘0) = (𝐹‘1) → (#‘𝐹) ≠ 2))
6250, 61sylbid 229 . . . . . . . . . . . . . . . . . . . 20 ((𝐸:dom 𝐸1-1→ran 𝐸𝐹:(0..^2)–1-1→dom 𝐸) → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2))
6362ex 449 . . . . . . . . . . . . . . . . . . 19 (𝐸:dom 𝐸1-1→ran 𝐸 → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2)))
6434, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (#‘𝐹) ≠ 2)))
6564com13 86 . . . . . . . . . . . . . . . . 17 ((𝐸‘(𝐹‘0)) = (𝐸‘(𝐹‘1)) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
6633, 65syl 17 . . . . . . . . . . . . . . . 16 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
6732, 66syl6bi 242 . . . . . . . . . . . . . . 15 ((𝑃‘0) = (𝑃‘2) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
6867com3l 87 . . . . . . . . . . . . . 14 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
6925, 68sylbi 206 . . . . . . . . . . . . 13 (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
7024, 69sylbi 206 . . . . . . . . . . . 12 (∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝐹:(0..^2)–1-1→dom 𝐸 → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))))
7170impcom 445 . . . . . . . . . . 11 ((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘2) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
7271imp31 447 . . . . . . . . . 10 ((((𝐹:(0..^2)–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^2)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘2)) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2)
7322, 72syl6bi 242 . . . . . . . . 9 ((#‘𝐹) = 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2))
74 ax-1 6 . . . . . . . . 9 ((#‘𝐹) ≠ 2 → ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2))
7573, 74pm2.61ine 2865 . . . . . . . 8 ((((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) ∧ 𝑉 USGrph 𝐸) → (#‘𝐹) ≠ 2)
7675exp31 628 . . . . . . 7 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
77763adant2 1073 . . . . . 6 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
7877adantld 482 . . . . 5 ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
7978adantl 481 . . . 4 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
8013, 79sylbid 229 . . 3 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2)))
8111, 80mpcom 37 . 2 (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → (#‘𝐹) ≠ 2))
8281impcom 445 1 ((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   Trails ctrail 26027   Paths cpath 26028   Cycles ccycl 26035 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-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-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-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-cycl 26041 This theorem is referenced by: (None)
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