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Theorem konigsberglem3 30051
Description: Lemma 3 for konigsberg 30054: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
Hypotheses
Ref Expression
konigsberg.v 𝑉 = (0...3)
konigsberg.e 𝐸 = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
konigsberg.g 𝐺 = βŸ¨π‘‰, 𝐸⟩
Assertion
Ref Expression
konigsberglem3 ((VtxDegβ€˜πΊ)β€˜3) = 3

Proof of Theorem konigsberglem3
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ovex 7447 . . . . 5 (0...3) ∈ V
2 s6cli 14859 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word V
32elexi 3489 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ V
41, 3opvtxfvi 28809 . . . 4 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = (0...3)
54eqcomi 2736 . . 3 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)
6 3nn0 12512 . . . 4 3 ∈ β„•0
7 nn0fz0 13623 . . . 4 (3 ∈ β„•0 ↔ 3 ∈ (0...3))
86, 7mpbi 229 . . 3 3 ∈ (0...3)
91, 3opiedgfvi 28810 . . . 4 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©
109eqcomi 2736 . . 3 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)
11 s1cli 14579 . . . 4 βŸ¨β€œ{2, 3}β€βŸ© ∈ Word V
12 df-s7 14828 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
13 eqid 2727 . . . . 5 (0...3) = (0...3)
14 eqid 2727 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
15 eqid 2727 . . . . 5 ⟨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©βŸ© = ⟨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©βŸ©
1613, 14, 15konigsbergssiedgw 30047 . . . 4 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
172, 11, 12, 16mp3an 1458 . . 3 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
18 s5cli 14858 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word V
1918elexi 3489 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ V
201, 19opvtxfvi 28809 . . . . . 6 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = (0...3)
2120eqcomi 2736 . . . . 5 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)
221, 19opiedgfvi 28810 . . . . . 6 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©
2322eqcomi 2736 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)
24 s2cli 14855 . . . . . 6 βŸ¨β€œ{2, 3} {2, 3}β€βŸ© ∈ Word V
25 s5s2 14910 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3} {2, 3}β€βŸ©)
2613, 14, 15konigsbergssiedgw 30047 . . . . . 6 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
2718, 24, 25, 26mp3an 1458 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
28 s4cli 14857 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word V
2928elexi 3489 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ V
301, 29opvtxfvi 28809 . . . . . . 7 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = (0...3)
3130eqcomi 2736 . . . . . 6 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)
321, 29opiedgfvi 28810 . . . . . . 7 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©
3332eqcomi 2736 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)
34 s3cli 14856 . . . . . . 7 βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
35 s4s3 14906 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ©)
3613, 14, 15konigsbergssiedgw 30047 . . . . . . 7 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
3728, 34, 35, 36mp3an 1458 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
38 s3cli 14856 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word V
3938elexi 3489 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ V
401, 39opvtxfvi 28809 . . . . . . . 8 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = (0...3)
4140eqcomi 2736 . . . . . . 7 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)
421, 39opiedgfvi 28810 . . . . . . . 8 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©
4342eqcomi 2736 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)
44 s4cli 14857 . . . . . . . 8 βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
45 s3s4 14908 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
4613, 14, 15konigsbergssiedgw 30047 . . . . . . . 8 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
4738, 44, 45, 46mp3an 1458 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
48 s2cli 14855 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word V
4948elexi 3489 . . . . . . . . . . 11 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ V
501, 49opvtxfvi 28809 . . . . . . . . . 10 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = (0...3)
5150eqcomi 2736 . . . . . . . . 9 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
521, 49opiedgfvi 28810 . . . . . . . . . 10 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2}β€βŸ©
5352eqcomi 2736 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
54 s5cli 14858 . . . . . . . . . 10 βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
55 s2s5 14909 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
5613, 14, 15konigsbergssiedgw 30047 . . . . . . . . . 10 ((βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
5748, 54, 55, 56mp3an 1458 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
58 s1cli 14579 . . . . . . . . . . . . 13 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word V
5958elexi 3489 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© ∈ V
601, 59opvtxfvi 28809 . . . . . . . . . . 11 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (0...3)
6160eqcomi 2736 . . . . . . . . . 10 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
621, 59opiedgfvi 28810 . . . . . . . . . . 11 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1}β€βŸ©
6362eqcomi 2736 . . . . . . . . . 10 βŸ¨β€œ{0, 1}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
64 s6cli 14859 . . . . . . . . . . 11 βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
65 s1s6 14902 . . . . . . . . . . 11 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
6613, 14, 15konigsbergssiedgw 30047 . . . . . . . . . . 11 ((βŸ¨β€œ{0, 1}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
6758, 64, 65, 66mp3an 1458 . . . . . . . . . 10 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
68 0ex 5301 . . . . . . . . . . . . 13 βˆ… ∈ V
691, 68opvtxfvi 28809 . . . . . . . . . . . 12 (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©) = (0...3)
7069eqcomi 2736 . . . . . . . . . . 11 (0...3) = (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©)
711, 68opiedgfvi 28810 . . . . . . . . . . . 12 (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©) = βˆ…
7271eqcomi 2736 . . . . . . . . . . 11 βˆ… = (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©)
73 wrd0 14513 . . . . . . . . . . 11 βˆ… ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
74 eqid 2727 . . . . . . . . . . . 12 βˆ… = βˆ…
7570, 72vtxdg0e 29275 . . . . . . . . . . . 12 ((3 ∈ (0...3) ∧ βˆ… = βˆ…) β†’ ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜3) = 0)
768, 74, 75mp2an 691 . . . . . . . . . . 11 ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜3) = 0
77 0elfz 13622 . . . . . . . . . . . 12 (3 ∈ β„•0 β†’ 0 ∈ (0...3))
786, 77ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
79 3ne0 12340 . . . . . . . . . . . 12 3 β‰  0
8079necomi 2990 . . . . . . . . . . 11 0 β‰  3
81 1nn0 12510 . . . . . . . . . . . 12 1 ∈ β„•0
82 1le3 12446 . . . . . . . . . . . 12 1 ≀ 3
83 elfz2nn0 13616 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 1 ≀ 3))
8481, 6, 82, 83mpbir3an 1339 . . . . . . . . . . 11 1 ∈ (0...3)
85 1re 11236 . . . . . . . . . . . 12 1 ∈ ℝ
86 1lt3 12407 . . . . . . . . . . . 12 1 < 3
8785, 86ltneii 11349 . . . . . . . . . . 11 1 β‰  3
88 s0s1 14897 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
8962, 88eqtri 2755 . . . . . . . . . . 11 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
9070, 8, 72, 73, 76, 60, 78, 80, 84, 87, 89vdegp1ai 29337 . . . . . . . . . 10 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)β€˜3) = 0
91 2nn0 12511 . . . . . . . . . . 11 2 ∈ β„•0
92 2re 12308 . . . . . . . . . . . 12 2 ∈ ℝ
93 3re 12314 . . . . . . . . . . . 12 3 ∈ ℝ
94 2lt3 12406 . . . . . . . . . . . 12 2 < 3
9592, 93, 94ltleii 11359 . . . . . . . . . . 11 2 ≀ 3
96 elfz2nn0 13616 . . . . . . . . . . 11 (2 ∈ (0...3) ↔ (2 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 2 ≀ 3))
9791, 6, 95, 96mpbir3an 1339 . . . . . . . . . 10 2 ∈ (0...3)
9892, 94ltneii 11349 . . . . . . . . . 10 2 β‰  3
99 df-s2 14823 . . . . . . . . . . 11 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2}β€βŸ©)
10052, 99eqtri 2755 . . . . . . . . . 10 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2}β€βŸ©)
10161, 8, 63, 67, 90, 50, 78, 80, 97, 98, 100vdegp1ai 29337 . . . . . . . . 9 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)β€˜3) = 0
102 df-s3 14824 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3}β€βŸ©)
10342, 102eqtri 2755 . . . . . . . . 9 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3}β€βŸ©)
10451, 8, 53, 57, 101, 40, 78, 80, 103vdegp1ci 29339 . . . . . . . 8 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)β€˜3) = (0 + 1)
105 0p1e1 12356 . . . . . . . 8 (0 + 1) = 1
106104, 105eqtri 2755 . . . . . . 7 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)β€˜3) = 1
107 df-s4 14825 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
10832, 107eqtri 2755 . . . . . . 7 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
10941, 8, 43, 47, 106, 30, 84, 87, 97, 98, 108vdegp1ai 29337 . . . . . 6 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)β€˜3) = 1
110 df-s5 14826 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
11122, 110eqtri 2755 . . . . . 6 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
11231, 8, 33, 37, 109, 20, 84, 87, 97, 98, 111vdegp1ai 29337 . . . . 5 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)β€˜3) = 1
113 df-s6 14827 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
1149, 113eqtri 2755 . . . . 5 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
11521, 8, 23, 27, 112, 4, 97, 98, 114vdegp1ci 29339 . . . 4 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)β€˜3) = (1 + 1)
116 1p1e2 12359 . . . 4 (1 + 1) = 2
117115, 116eqtri 2755 . . 3 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)β€˜3) = 2
118 konigsberg.v . . . 4 𝑉 = (0...3)
119 konigsberg.e . . . 4 𝐸 = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
120 konigsberg.g . . . 4 𝐺 = βŸ¨π‘‰, 𝐸⟩
121118, 119, 120konigsbergvtx 30043 . . 3 (Vtxβ€˜πΊ) = (0...3)
122118, 119, 120konigsbergiedg 30044 . . . 4 (iEdgβ€˜πΊ) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
123122, 12eqtri 2755 . . 3 (iEdgβ€˜πΊ) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
1245, 8, 10, 17, 117, 121, 97, 98, 123vdegp1ci 29339 . 2 ((VtxDegβ€˜πΊ)β€˜3) = (2 + 1)
125 2p1e3 12376 . 2 (2 + 1) = 3
126124, 125eqtri 2755 1 ((VtxDegβ€˜πΊ)β€˜3) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   βˆ– cdif 3941  βˆ…c0 4318  π’« cpw 4598  {csn 4624  {cpr 4626  βŸ¨cop 4630   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131   + caddc 11133   ≀ cle 11271  2c2 12289  3c3 12290  β„•0cn0 12494  ...cfz 13508  β™―chash 14313  Word cword 14488   ++ cconcat 14544  βŸ¨β€œcs1 14569  βŸ¨β€œcs2 14816  βŸ¨β€œcs3 14817  βŸ¨β€œcs4 14818  βŸ¨β€œcs5 14819  βŸ¨β€œcs6 14820  βŸ¨β€œcs7 14821  Vtxcvtx 28796  iEdgciedg 28797  VtxDegcvtxdg 29266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-xadd 13117  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-s4 14825  df-s5 14826  df-s6 14827  df-s7 14828  df-vtx 28798  df-iedg 28799  df-vtxdg 29267
This theorem is referenced by:  konigsberglem4  30052
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