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Theorem konigsberglem2 29200
Description: Lemma 2 for konigsberg 29204: Vertex 1 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
konigsberglem2 ((VtxDegβ€˜πΊ)β€˜1) = 3

Proof of Theorem konigsberglem2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ovex 7391 . . . 4 (0...3) ∈ V
2 s6cli 14774 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word V
32elexi 3465 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ V
41, 3opvtxfvi 27963 . . 3 (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 2746 . 2 (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 1nn0 12430 . . 3 1 ∈ β„•0
7 3nn0 12432 . . 3 3 ∈ β„•0
8 1le3 12366 . . 3 1 ≀ 3
9 elfz2nn0 13533 . . 3 (1 ∈ (0...3) ↔ (1 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 1 ≀ 3))
106, 7, 8, 9mpbir3an 1342 . 2 1 ∈ (0...3)
111, 3opiedgfvi 27964 . . 3 (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©
1211eqcomi 2746 . 2 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©)
13 s1cli 14494 . . 3 βŸ¨β€œ{2, 3}β€βŸ© ∈ Word V
14 df-s7 14743 . . 3 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©)
15 eqid 2737 . . . 4 (0...3) = (0...3)
16 eqid 2737 . . . 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} {2, 3}β€βŸ©
17 eqid 2737 . . . 4 ⟨(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©
1815, 16, 17konigsbergssiedgw 29197 . . 3 ((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})
192, 13, 14, 18mp3an 1462 . 2 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}
20 s5cli 14773 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word V
2120elexi 3465 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ V
221, 21opvtxfvi 27963 . . . 4 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = (0...3)
2322eqcomi 2746 . . 3 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)
241, 21opiedgfvi 27964 . . . 4 (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©
2524eqcomi 2746 . . 3 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©)
26 s2cli 14770 . . . 4 βŸ¨β€œ{2, 3} {2, 3}β€βŸ© ∈ Word V
27 s5s2 14825 . . . 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}β€βŸ© ++ βŸ¨β€œ{2, 3} {2, 3}β€βŸ©)
2815, 16, 17konigsbergssiedgw 29197 . . . 4 ((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})
2920, 26, 27, 28mp3an 1462 . . 3 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}
30 s4cli 14772 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word V
3130elexi 3465 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ V
321, 31opvtxfvi 27963 . . . . . 6 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = (0...3)
3332eqcomi 2746 . . . . 5 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)
341, 31opiedgfvi 27964 . . . . . 6 (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©
3534eqcomi 2746 . . . . 5 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©)
36 s3cli 14771 . . . . . 6 βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
37 s4s3 14821 . . . . . 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}β€βŸ© ++ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ©)
3815, 16, 17konigsbergssiedgw 29197 . . . . . 6 ((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})
3930, 36, 37, 38mp3an 1462 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
40 s3cli 14771 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word V
4140elexi 3465 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ V
421, 41opvtxfvi 27963 . . . . . . . 8 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = (0...3)
4342eqcomi 2746 . . . . . . 7 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)
441, 41opiedgfvi 27964 . . . . . . . 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©
4544eqcomi 2746 . . . . . . 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©)
46 s4cli 14772 . . . . . . . 8 βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
47 s3s4 14823 . . . . . . . 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©)
4815, 16, 17konigsbergssiedgw 29197 . . . . . . . 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})
4940, 46, 47, 48mp3an 1462 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
50 s2cli 14770 . . . . . . . . . . 11 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word V
5150elexi 3465 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ V
521, 51opvtxfvi 27963 . . . . . . . . 9 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = (0...3)
5352eqcomi 2746 . . . . . . . 8 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
541, 51opiedgfvi 27964 . . . . . . . . 9 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2}β€βŸ©
5554eqcomi 2746 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
56 s5cli 14773 . . . . . . . . 9 βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
57 s2s5 14824 . . . . . . . . 9 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©)
5815, 16, 17konigsbergssiedgw 29197 . . . . . . . . 9 ((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})
5950, 56, 57, 58mp3an 1462 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
60 s1cli 14494 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word V
6160elexi 3465 . . . . . . . . . . 11 βŸ¨β€œ{0, 1}β€βŸ© ∈ V
621, 61opvtxfvi 27963 . . . . . . . . . 10 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (0...3)
6362eqcomi 2746 . . . . . . . . 9 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
641, 61opiedgfvi 27964 . . . . . . . . . 10 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1}β€βŸ©
6564eqcomi 2746 . . . . . . . . 9 βŸ¨β€œ{0, 1}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
66 s6cli 14774 . . . . . . . . . 10 βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
67 s1s6 14817 . . . . . . . . . 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}β€βŸ© ++ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
6815, 16, 17konigsbergssiedgw 29197 . . . . . . . . . 10 ((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})
6960, 66, 67, 68mp3an 1462 . . . . . . . . 9 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
70 0ex 5265 . . . . . . . . . . . . 13 βˆ… ∈ V
711, 70opvtxfvi 27963 . . . . . . . . . . . 12 (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©) = (0...3)
7271eqcomi 2746 . . . . . . . . . . 11 (0...3) = (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©)
731, 70opiedgfvi 27964 . . . . . . . . . . . 12 (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©) = βˆ…
7473eqcomi 2746 . . . . . . . . . . 11 βˆ… = (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©)
75 wrd0 14428 . . . . . . . . . . 11 βˆ… ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
76 eqid 2737 . . . . . . . . . . . 12 βˆ… = βˆ…
7772, 74vtxdg0e 28425 . . . . . . . . . . . 12 ((1 ∈ (0...3) ∧ βˆ… = βˆ…) β†’ ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜1) = 0)
7810, 76, 77mp2an 691 . . . . . . . . . . 11 ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜1) = 0
79 0elfz 13539 . . . . . . . . . . . 12 (3 ∈ β„•0 β†’ 0 ∈ (0...3))
807, 79ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
81 0ne1 12225 . . . . . . . . . . 11 0 β‰  1
82 s0s1 14812 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
8364, 82eqtri 2765 . . . . . . . . . . 11 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
8472, 10, 74, 75, 78, 62, 80, 81, 83vdegp1ci 28489 . . . . . . . . . 10 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)β€˜1) = (0 + 1)
85 0p1e1 12276 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2765 . . . . . . . . 9 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)β€˜1) = 1
87 2nn0 12431 . . . . . . . . . 10 2 ∈ β„•0
88 2re 12228 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 12234 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 12326 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 11279 . . . . . . . . . 10 2 ≀ 3
92 elfz2nn0 13533 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 2 ≀ 3))
9387, 7, 91, 92mpbir3an 1342 . . . . . . . . 9 2 ∈ (0...3)
94 1ne2 12362 . . . . . . . . . 10 1 β‰  2
9594necomi 2999 . . . . . . . . 9 2 β‰  1
96 df-s2 14738 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2}β€βŸ©)
9754, 96eqtri 2765 . . . . . . . . 9 (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©)
9863, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97vdegp1ai 28487 . . . . . . . 8 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)β€˜1) = 1
99 nn0fz0 13540 . . . . . . . . 9 (3 ∈ β„•0 ↔ 3 ∈ (0...3))
1007, 99mpbi 229 . . . . . . . 8 3 ∈ (0...3)
101 1re 11156 . . . . . . . . 9 1 ∈ ℝ
102 1lt3 12327 . . . . . . . . 9 1 < 3
103101, 102gtneii 11268 . . . . . . . 8 3 β‰  1
104 df-s3 14739 . . . . . . . . 9 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©)
10544, 104eqtri 2765 . . . . . . . 8 (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©)
10653, 10, 55, 59, 98, 42, 80, 81, 100, 103, 105vdegp1ai 28487 . . . . . . 7 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)β€˜1) = 1
107 df-s4 14740 . . . . . . . 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©)
10834, 107eqtri 2765 . . . . . . 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©)
10943, 10, 45, 49, 106, 32, 93, 95, 108vdegp1bi 28488 . . . . . 6 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)β€˜1) = (1 + 1)
110 1p1e2 12279 . . . . . 6 (1 + 1) = 2
111109, 110eqtri 2765 . . . . 5 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)β€˜1) = 2
112 df-s5 14741 . . . . . 6 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©)
11324, 112eqtri 2765 . . . . 5 (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©)
11433, 10, 35, 39, 111, 22, 93, 95, 113vdegp1bi 28488 . . . 4 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)β€˜1) = (2 + 1)
115 2p1e3 12296 . . . 4 (2 + 1) = 3
116114, 115eqtri 2765 . . 3 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)β€˜1) = 3
117 df-s6 14742 . . . 4 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©)
11811, 117eqtri 2765 . . 3 (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©)
11923, 10, 25, 29, 116, 4, 93, 95, 100, 103, 118vdegp1ai 28487 . 2 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)β€˜1) = 3
120 konigsberg.v . . 3 𝑉 = (0...3)
121 konigsberg.e . . 3 𝐸 = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
122 konigsberg.g . . 3 𝐺 = βŸ¨π‘‰, 𝐸⟩
123120, 121, 122konigsbergvtx 29193 . 2 (Vtxβ€˜πΊ) = (0...3)
124120, 121, 122konigsbergiedg 29194 . . 3 (iEdgβ€˜πΊ) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
125124, 14eqtri 2765 . 2 (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©)
1265, 10, 12, 19, 119, 123, 93, 95, 100, 103, 125vdegp1ai 28487 1 ((VtxDegβ€˜πΊ)β€˜1) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  {crab 3408  Vcvv 3446   βˆ– cdif 3908  βˆ…c0 4283  π’« cpw 4561  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  0cc0 11052  1c1 11053   + caddc 11055   ≀ cle 11191  2c2 12209  3c3 12210  β„•0cn0 12414  ...cfz 13425  β™―chash 14231  Word cword 14403   ++ cconcat 14459  βŸ¨β€œcs1 14484  βŸ¨β€œcs2 14731  βŸ¨β€œcs3 14732  βŸ¨β€œcs4 14733  βŸ¨β€œcs5 14734  βŸ¨β€œcs6 14735  βŸ¨β€œcs7 14736  Vtxcvtx 27950  iEdgciedg 27951  VtxDegcvtxdg 28416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-xnn0 12487  df-z 12501  df-uz 12765  df-xadd 13035  df-fz 13426  df-fzo 13569  df-hash 14232  df-word 14404  df-concat 14460  df-s1 14485  df-s2 14738  df-s3 14739  df-s4 14740  df-s5 14741  df-s6 14742  df-s7 14743  df-vtx 27952  df-iedg 27953  df-vtxdg 28417
This theorem is referenced by:  konigsberglem4  29202
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