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Theorem konigsberg 26514
 Description: The Konigsberg Bridge problem. If ⟨𝑉, 𝐸⟩ is the graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eupath 26508 the graph cannot have an Eulerian path. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
konigsberg.v 𝑉 = (0...3)
konigsberg.e 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
Assertion
Ref Expression
konigsberg (𝑉 EulPaths 𝐸) = ∅

Proof of Theorem konigsberg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 prhash2ex 13048 . . . . . 6 (#‘{0, 1}) = 2
21oveq1i 6559 . . . . 5 ((#‘{0, 1}) + 1) = (2 + 1)
3 prfi 8120 . . . . . 6 {0, 1} ∈ Fin
4 3ne0 10992 . . . . . . 7 3 ≠ 0
5 1re 9918 . . . . . . . 8 1 ∈ ℝ
6 1lt3 11073 . . . . . . . 8 1 < 3
75, 6gtneii 10028 . . . . . . 7 3 ≠ 1
84, 7nelpri 4149 . . . . . 6 ¬ 3 ∈ {0, 1}
9 3nn0 11187 . . . . . . 7 3 ∈ ℕ0
10 hashunsng 13042 . . . . . . 7 (3 ∈ ℕ0 → (({0, 1} ∈ Fin ∧ ¬ 3 ∈ {0, 1}) → (#‘({0, 1} ∪ {3})) = ((#‘{0, 1}) + 1)))
119, 10ax-mp 5 . . . . . 6 (({0, 1} ∈ Fin ∧ ¬ 3 ∈ {0, 1}) → (#‘({0, 1} ∪ {3})) = ((#‘{0, 1}) + 1))
123, 8, 11mp2an 704 . . . . 5 (#‘({0, 1} ∪ {3})) = ((#‘{0, 1}) + 1)
13 df-3 10957 . . . . 5 3 = (2 + 1)
142, 12, 133eqtr4i 2642 . . . 4 (#‘({0, 1} ∪ {3})) = 3
15 konigsberg.v . . . . . . . 8 𝑉 = (0...3)
16 fzfi 12633 . . . . . . . 8 (0...3) ∈ Fin
1715, 16eqeltri 2684 . . . . . . 7 𝑉 ∈ Fin
18 ssrab2 3650 . . . . . . 7 {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ⊆ 𝑉
19 ssfi 8065 . . . . . . 7 ((𝑉 ∈ Fin ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ⊆ 𝑉) → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∈ Fin)
2017, 18, 19mp2an 704 . . . . . 6 {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∈ Fin
21 nn0uz 11598 . . . . . . . . . . . 12 0 = (ℤ‘0)
229, 21eleqtri 2686 . . . . . . . . . . 11 3 ∈ (ℤ‘0)
23 eluzfz1 12219 . . . . . . . . . . 11 (3 ∈ (ℤ‘0) → 0 ∈ (0...3))
2422, 23ax-mp 5 . . . . . . . . . 10 0 ∈ (0...3)
2524, 15eleqtrri 2687 . . . . . . . . 9 0 ∈ 𝑉
26 n2dvds3 14945 . . . . . . . . 9 ¬ 2 ∥ 3
27 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘0))
2815ovexi 6578 . . . . . . . . . . . . . 14 𝑉 ∈ V
29 df-s6 13448 . . . . . . . . . . . . . . 15 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
30 df-s5 13447 . . . . . . . . . . . . . . . 16 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
31 df-s4 13446 . . . . . . . . . . . . . . . . 17 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
32 df-s3 13445 . . . . . . . . . . . . . . . . . 18 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
33 df-s2 13444 . . . . . . . . . . . . . . . . . . 19 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
34 1le3 11121 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 3
35 1eluzge0 11608 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ (ℤ‘0)
36 3z 11287 . . . . . . . . . . . . . . . . . . . . . . . 24 3 ∈ ℤ
37 elfz5 12205 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1 ∈ (ℤ‘0) ∧ 3 ∈ ℤ) → (1 ∈ (0...3) ↔ 1 ≤ 3))
3835, 36, 37mp2an 704 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ (0...3) ↔ 1 ≤ 3)
3934, 38mpbir 220 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ (0...3)
4039, 15eleqtrri 2687 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ 𝑉
4128, 25, 40umgrabi 26510 . . . . . . . . . . . . . . . . . . . 20 (⊤ → {0, 1} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4241s1cld 13236 . . . . . . . . . . . . . . . . . . 19 (⊤ → ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
43 2re 10967 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ ℝ
44 3re 10971 . . . . . . . . . . . . . . . . . . . . . . 23 3 ∈ ℝ
45 2lt3 11072 . . . . . . . . . . . . . . . . . . . . . . 23 2 < 3
4643, 44, 45ltleii 10039 . . . . . . . . . . . . . . . . . . . . . 22 2 ≤ 3
47 2eluzge0 11609 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ (ℤ‘0)
48 elfz5 12205 . . . . . . . . . . . . . . . . . . . . . . 23 ((2 ∈ (ℤ‘0) ∧ 3 ∈ ℤ) → (2 ∈ (0...3) ↔ 2 ≤ 3))
4947, 36, 48mp2an 704 . . . . . . . . . . . . . . . . . . . . . 22 (2 ∈ (0...3) ↔ 2 ≤ 3)
5046, 49mpbir 220 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ (0...3)
5150, 15eleqtrri 2687 . . . . . . . . . . . . . . . . . . . 20 2 ∈ 𝑉
5228, 25, 51umgrabi 26510 . . . . . . . . . . . . . . . . . . 19 (⊤ → {0, 2} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5333, 42, 52cats1cld 13451 . . . . . . . . . . . . . . . . . 18 (⊤ → ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
54 eluzfz2 12220 . . . . . . . . . . . . . . . . . . . . 21 (3 ∈ (ℤ‘0) → 3 ∈ (0...3))
5522, 54ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 3 ∈ (0...3)
5655, 15eleqtrri 2687 . . . . . . . . . . . . . . . . . . 19 3 ∈ 𝑉
5728, 25, 56umgrabi 26510 . . . . . . . . . . . . . . . . . 18 (⊤ → {0, 3} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5832, 53, 57cats1cld 13451 . . . . . . . . . . . . . . . . 17 (⊤ → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5928, 40, 51umgrabi 26510 . . . . . . . . . . . . . . . . 17 (⊤ → {1, 2} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6031, 58, 59cats1cld 13451 . . . . . . . . . . . . . . . 16 (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6130, 60, 59cats1cld 13451 . . . . . . . . . . . . . . 15 (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6228, 51, 56umgrabi 26510 . . . . . . . . . . . . . . 15 (⊤ → {2, 3} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6329, 61, 62cats1cld 13451 . . . . . . . . . . . . . 14 (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
64 wrd0 13185 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
6564a1i 11 . . . . . . . . . . . . . . . . . . . 20 (⊤ → ∅ ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6628, 25vdeg0i 26509 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 VDeg ∅)‘0) = 0
67 1e0p1 11428 . . . . . . . . . . . . . . . . . . . 20 1 = (0 + 1)
68 ax-1ne0 9884 . . . . . . . . . . . . . . . . . . . 20 1 ≠ 0
69 s0s1 13517 . . . . . . . . . . . . . . . . . . . 20 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
7028, 65, 25, 66, 67, 40, 68, 69vdegp1bi 26512 . . . . . . . . . . . . . . . . . . 19 ((𝑉 VDeg ⟨“{0, 1}”⟩)‘0) = 1
71 df-2 10956 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
72 2ne0 10990 . . . . . . . . . . . . . . . . . . 19 2 ≠ 0
7328, 42, 25, 70, 71, 51, 72, 33vdegp1bi 26512 . . . . . . . . . . . . . . . . . 18 ((𝑉 VDeg ⟨“{0, 1} {0, 2}”⟩)‘0) = 2
7428, 53, 25, 73, 13, 56, 4, 32vdegp1bi 26512 . . . . . . . . . . . . . . . . 17 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3}”⟩)‘0) = 3
7528, 58, 25, 74, 40, 68, 51, 72, 31vdegp1ai 26511 . . . . . . . . . . . . . . . 16 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩)‘0) = 3
7628, 60, 25, 75, 40, 68, 51, 72, 30vdegp1ai 26511 . . . . . . . . . . . . . . 15 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩)‘0) = 3
7728, 61, 25, 76, 51, 72, 56, 4, 29vdegp1ai 26511 . . . . . . . . . . . . . 14 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩)‘0) = 3
78 konigsberg.e . . . . . . . . . . . . . . 15 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
79 df-s7 13449 . . . . . . . . . . . . . . 15 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
8078, 79eqtri 2632 . . . . . . . . . . . . . 14 𝐸 = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
8128, 63, 25, 77, 51, 72, 56, 4, 80vdegp1ai 26511 . . . . . . . . . . . . 13 ((𝑉 VDeg 𝐸)‘0) = 3
8227, 81syl6eq 2660 . . . . . . . . . . . 12 (𝑥 = 0 → ((𝑉 VDeg 𝐸)‘𝑥) = 3)
8382breq2d 4595 . . . . . . . . . . 11 (𝑥 = 0 → (2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ 2 ∥ 3))
8483notbid 307 . . . . . . . . . 10 (𝑥 = 0 → (¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ ¬ 2 ∥ 3))
8584elrab 3331 . . . . . . . . 9 (0 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ↔ (0 ∈ 𝑉 ∧ ¬ 2 ∥ 3))
8625, 26, 85mpbir2an 957 . . . . . . . 8 0 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
87 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 1 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘1))
8828, 40vdeg0i 26509 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 VDeg ∅)‘1) = 0
89 0ne1 10965 . . . . . . . . . . . . . . . . . . . 20 0 ≠ 1
9028, 65, 40, 88, 67, 25, 89, 69vdegp1ci 26513 . . . . . . . . . . . . . . . . . . 19 ((𝑉 VDeg ⟨“{0, 1}”⟩)‘1) = 1
91 1lt2 11071 . . . . . . . . . . . . . . . . . . . 20 1 < 2
925, 91gtneii 10028 . . . . . . . . . . . . . . . . . . 19 2 ≠ 1
9328, 42, 40, 90, 25, 89, 51, 92, 33vdegp1ai 26511 . . . . . . . . . . . . . . . . . 18 ((𝑉 VDeg ⟨“{0, 1} {0, 2}”⟩)‘1) = 1
9428, 53, 40, 93, 25, 89, 56, 7, 32vdegp1ai 26511 . . . . . . . . . . . . . . . . 17 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3}”⟩)‘1) = 1
9528, 58, 40, 94, 71, 51, 92, 31vdegp1bi 26512 . . . . . . . . . . . . . . . 16 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩)‘1) = 2
9628, 60, 40, 95, 13, 51, 92, 30vdegp1bi 26512 . . . . . . . . . . . . . . 15 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩)‘1) = 3
9728, 61, 40, 96, 51, 92, 56, 7, 29vdegp1ai 26511 . . . . . . . . . . . . . 14 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩)‘1) = 3
9828, 63, 40, 97, 51, 92, 56, 7, 80vdegp1ai 26511 . . . . . . . . . . . . 13 ((𝑉 VDeg 𝐸)‘1) = 3
9987, 98syl6eq 2660 . . . . . . . . . . . 12 (𝑥 = 1 → ((𝑉 VDeg 𝐸)‘𝑥) = 3)
10099breq2d 4595 . . . . . . . . . . 11 (𝑥 = 1 → (2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ 2 ∥ 3))
101100notbid 307 . . . . . . . . . 10 (𝑥 = 1 → (¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ ¬ 2 ∥ 3))
102101elrab 3331 . . . . . . . . 9 (1 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ↔ (1 ∈ 𝑉 ∧ ¬ 2 ∥ 3))
10340, 26, 102mpbir2an 957 . . . . . . . 8 1 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
104 prssi 4293 . . . . . . . 8 ((0 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∧ 1 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) → {0, 1} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
10586, 103, 104mp2an 704 . . . . . . 7 {0, 1} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
106 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 3 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘3))
10728, 56vdeg0i 26509 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 VDeg ∅)‘3) = 0
108 0re 9919 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ
109 3pos 10991 . . . . . . . . . . . . . . . . . . . . 21 0 < 3
110108, 109ltneii 10029 . . . . . . . . . . . . . . . . . . . 20 0 ≠ 3
1115, 6ltneii 10029 . . . . . . . . . . . . . . . . . . . 20 1 ≠ 3
11228, 65, 56, 107, 25, 110, 40, 111, 69vdegp1ai 26511 . . . . . . . . . . . . . . . . . . 19 ((𝑉 VDeg ⟨“{0, 1}”⟩)‘3) = 0
11343, 45ltneii 10029 . . . . . . . . . . . . . . . . . . 19 2 ≠ 3
11428, 42, 56, 112, 25, 110, 51, 113, 33vdegp1ai 26511 . . . . . . . . . . . . . . . . . 18 ((𝑉 VDeg ⟨“{0, 1} {0, 2}”⟩)‘3) = 0
11528, 53, 56, 114, 67, 25, 110, 32vdegp1ci 26513 . . . . . . . . . . . . . . . . 17 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3}”⟩)‘3) = 1
11628, 58, 56, 115, 40, 111, 51, 113, 31vdegp1ai 26511 . . . . . . . . . . . . . . . 16 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩)‘3) = 1
11728, 60, 56, 116, 40, 111, 51, 113, 30vdegp1ai 26511 . . . . . . . . . . . . . . 15 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩)‘3) = 1
11828, 61, 56, 117, 71, 51, 113, 29vdegp1ci 26513 . . . . . . . . . . . . . 14 ((𝑉 VDeg ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩)‘3) = 2
11928, 63, 56, 118, 13, 51, 113, 80vdegp1ci 26513 . . . . . . . . . . . . 13 ((𝑉 VDeg 𝐸)‘3) = 3
120106, 119syl6eq 2660 . . . . . . . . . . . 12 (𝑥 = 3 → ((𝑉 VDeg 𝐸)‘𝑥) = 3)
121120breq2d 4595 . . . . . . . . . . 11 (𝑥 = 3 → (2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ 2 ∥ 3))
122121notbid 307 . . . . . . . . . 10 (𝑥 = 3 → (¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥) ↔ ¬ 2 ∥ 3))
123122elrab 3331 . . . . . . . . 9 (3 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ↔ (3 ∈ 𝑉 ∧ ¬ 2 ∥ 3))
12456, 26, 123mpbir2an 957 . . . . . . . 8 3 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
125 snssi 4280 . . . . . . . 8 (3 ∈ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} → {3} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
126124, 125ax-mp 5 . . . . . . 7 {3} ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
127105, 126unssi 3750 . . . . . 6 ({0, 1} ∪ {3}) ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
128 ssdomg 7887 . . . . . 6 ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∈ Fin → (({0, 1} ∪ {3}) ⊆ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} → ({0, 1} ∪ {3}) ≼ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}))
12920, 127, 128mp2 9 . . . . 5 ({0, 1} ∪ {3}) ≼ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}
130 snfi 7923 . . . . . . 7 {3} ∈ Fin
131 unfi 8112 . . . . . . 7 (({0, 1} ∈ Fin ∧ {3} ∈ Fin) → ({0, 1} ∪ {3}) ∈ Fin)
1323, 130, 131mp2an 704 . . . . . 6 ({0, 1} ∪ {3}) ∈ Fin
133 hashdom 13029 . . . . . 6 ((({0, 1} ∪ {3}) ∈ Fin ∧ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∈ Fin) → ((#‘({0, 1} ∪ {3})) ≤ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ↔ ({0, 1} ∪ {3}) ≼ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}))
134132, 20, 133mp2an 704 . . . . 5 ((#‘({0, 1} ∪ {3})) ≤ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ↔ ({0, 1} ∪ {3}) ≼ {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
135129, 134mpbir 220 . . . 4 (#‘({0, 1} ∪ {3})) ≤ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
13614, 135eqbrtrri 4606 . . 3 3 ≤ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)})
137 hashcl 13009 . . . . . 6 ({𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} ∈ Fin → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ ℕ0)
13820, 137ax-mp 5 . . . . 5 (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ ℕ0
139138nn0rei 11180 . . . 4 (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ ℝ
14044, 139lenlti 10036 . . 3 (3 ≤ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ↔ ¬ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3)
141136, 140mpbi 219 . 2 ¬ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3
142 eupath 26508 . . . 4 ((𝑉 EulPaths 𝐸) ≠ ∅ → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ {0, 2})
143 elpri 4145 . . . 4 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ {0, 2} → ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 0 ∨ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 2))
144 id 22 . . . . . 6 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 0 → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 0)
145144, 109syl6eqbr 4622 . . . . 5 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 0 → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3)
146 id 22 . . . . . 6 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 2 → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 2)
147146, 45syl6eqbr 4622 . . . . 5 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 2 → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3)
148145, 147jaoi 393 . . . 4 (((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 0 ∨ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) = 2) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3)
149142, 143, 1483syl 18 . . 3 ((𝑉 EulPaths 𝐸) ≠ ∅ → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3)
150149necon1bi 2810 . 2 (¬ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) < 3 → (𝑉 EulPaths 𝐸) = ∅)
151141, 150ax-mp 5 1 (𝑉 EulPaths 𝐸) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ≼ cdom 7839  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  2c2 10947  3c3 10948  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149  ⟨“cs2 13437  ⟨“cs3 13438  ⟨“cs4 13439  ⟨“cs5 13440  ⟨“cs6 13441  ⟨“cs7 13442   ∥ cdvds 14821   VDeg cvdg 26420   EulPaths ceup 26489 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  ax-pre-sup 9893 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-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  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-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-s4 13446  df-s5 13447  df-s6 13448  df-s7 13449  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-umgra 25842  df-vdgr 26421  df-eupa 26490 This theorem is referenced by: (None)
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