Theorem gpg5nbgr3star 48557

Description: In a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), these are the Petersen graph G(5,2) and the 5-prism G(5,1), every vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every vertex induces a subgraph which is isomorphic to a 3-star). This does not hold for every generalized Petersen graph: for example, in the 3-prism G(3,1) (see gpg31grim3prism TODO) and the Dürer graph G(6,2) there are vertices which have neighborhoods containing triangles. In general, all generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles, see gpg3kgrtriex 48565. (Contributed by AV, 8-Sep-2025.)

Hypotheses

Ref	Expression
gpgnbgr.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgnbgr.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)
gpgnbgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
gpgnbgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
gpg5nbgr3star	⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))

Distinct variable groups:   𝑦,𝐺   𝑦,𝑉   𝑦,𝑋   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑁,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉   𝑥,𝑋   𝑥,𝐸,𝑦

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem gpg5nbgr3star

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		5eluz3 12833	. . . . . 6 ⊢ 5 ∈ (ℤ≥‘3)
2		eleq1 2824	. . . . . 6 ⊢ (𝑁 = 5 → (𝑁 ∈ (ℤ≥‘3) ↔ 5 ∈ (ℤ≥‘3)))
3	1, 2	mpbiri 258	. . . . 5 ⊢ (𝑁 = 5 → 𝑁 ∈ (ℤ≥‘3))
4	3	anim1i 616	. . . 4 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
5		eqid 2736	. . . . 5 ⊢ (0..^𝑁) = (0..^𝑁)
6		gpgnbgr.j	. . . . 5 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
7		gpgnbgr.g	. . . . 5 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
8		gpgnbgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
9	5, 6, 7, 8	gpgvtxel 48523	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑎 ∈ {0, 1}∃𝑏 ∈ (0..^𝑁)𝑋 = ⟨𝑎, 𝑏⟩))
10	4, 9	syl 17	. . 3 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑎 ∈ {0, 1}∃𝑏 ∈ (0..^𝑁)𝑋 = ⟨𝑎, 𝑏⟩))
11	10	biimp3a 1472	. 2 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ∃𝑎 ∈ {0, 1}∃𝑏 ∈ (0..^𝑁)𝑋 = ⟨𝑎, 𝑏⟩)
12		elpri 4591	. . . . . . 7 ⊢ (𝑎 ∈ {0, 1} → (𝑎 = 0 ∨ 𝑎 = 1))
13		opeq1 4816	. . . . . . . . . . . 12 ⊢ (𝑎 = 0 → ⟨𝑎, 𝑏⟩ = ⟨0, 𝑏⟩)
14	13	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑎 = 0 → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨0, 𝑏⟩))
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝑎 = 0 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨0, 𝑏⟩))
16		c0ex 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
17		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
18	16, 17	op1std 7952	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨0, 𝑏⟩ → (1st ‘𝑋) = 0)
19		4z 12561	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℤ
20		5nn 12267	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℕ
21	20	nnzi 12551	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℤ
22		4re 12265	. . . . . . . . . . . . . . . . . 18 ⊢ 4 ∈ ℝ
23		5re 12268	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℝ
24		4lt5 12353	. . . . . . . . . . . . . . . . . 18 ⊢ 4 < 5
25	22, 23, 24	ltleii 11269	. . . . . . . . . . . . . . . . 17 ⊢ 4 ≤ 5
26		eluz2 12794	. . . . . . . . . . . . . . . . 17 ⊢ (5 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 5 ∈ ℤ ∧ 4 ≤ 5))
27	19, 21, 25, 26	mpbir3an 1343	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ (ℤ≥‘4)
28		eleq1 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = 5 → (𝑁 ∈ (ℤ≥‘4) ↔ 5 ∈ (ℤ≥‘4)))
29	27, 28	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = 5 → 𝑁 ∈ (ℤ≥‘4))
30		gpgnbgr.u	. . . . . . . . . . . . . . . 16 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
31		gpgnbgr.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (Edg‘𝐺)
32	6, 7, 8, 30, 31	gpg5nbgrvtx03star 48556	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))
33	29, 32	sylanl1 681	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))
34	33	exp43 436	. . . . . . . . . . . . 13 ⊢ (𝑁 = 5 → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 0 → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))))
35	34	3imp 1111	. . . . . . . . . . . 12 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 0 → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
36	18, 35	syl5 34	. . . . . . . . . . 11 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨0, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝑎 = 0 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨0, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
38	15, 37	sylbid 240	. . . . . . . . 9 ⊢ ((𝑎 = 0 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
39	38	ex 412	. . . . . . . 8 ⊢ (𝑎 = 0 → ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
40		opeq1 4816	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → ⟨𝑎, 𝑏⟩ = ⟨1, 𝑏⟩)
41	40	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨1, 𝑏⟩))
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨1, 𝑏⟩))
43		1ex 11140	. . . . . . . . . . . . 13 ⊢ 1 ∈ V
44	43, 17	op1std 7952	. . . . . . . . . . . 12 ⊢ (𝑋 = ⟨1, 𝑏⟩ → (1st ‘𝑋) = 1)
45	6, 7, 8, 30	gpg3nbgrvtx1 48554	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
46	3, 45	sylanl1 681	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3)
47		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩
48	6	eleq2i 2828	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
49	48	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
50		gpgusgra 48533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
51	7, 50	eqeltrid 2840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
52	3, 49, 51	syl2an 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝐺 ∈ USGraph)
54	31	usgredgne 29275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸) → ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
55	54	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸) → ¬ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩)
56	55	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∈ USGraph → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩))
57	53, 56	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩))
58	47, 57	mt2i 137	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ¬ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸)
59		df-nel 3037	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ ¬ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∈ 𝐸)
60	58, 59	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸)
61		fvexd 6855	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → (2nd ‘𝑋) ∈ V)
62	6, 7, 8, 31	gpg5nbgrvtx13starlem1 48547	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (2nd ‘𝑋) ∈ V) → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸)
63	61, 62	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸)
64		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1) → 𝑋 ∈ 𝑉)
65	4, 64	anim12i 614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
66	5, 6, 7, 8	gpgvtxel2 48524	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
67		elfzoelz 13613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑋) ∈ (0..^𝑁) → (2nd ‘𝑋) ∈ ℤ)
68	65, 66, 67	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (2nd ‘𝑋) ∈ ℤ)
69	6, 7, 8, 31	gpg5nbgrvtx13starlem2 48548	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (2nd ‘𝑋) ∈ ℤ) → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
70	68, 69	syldan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
71		opex 5416	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ ∈ V
72		opex 5416	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨0, (2nd ‘𝑋)⟩ ∈ V
73		opex 5416	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ∈ V
74		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩})
75		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
76	74, 75	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
77		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩})
78		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
80		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
81		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
83	71, 72, 73, 76, 79, 82	raltp 4649	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
84	60, 63, 70, 83	syl3anbrc 1345	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸)
85		prcom 4676	. . . . . . . . . . . . . . . . . . . 20 ⊢ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩}
86		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} → ({⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
87	85, 86	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸)
88	63, 87	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸)
89		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨0, (2nd ‘𝑋)⟩ = ⟨0, (2nd ‘𝑋)⟩
90	31	usgredgne 29275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨0, (2nd ‘𝑋)⟩ ≠ ⟨0, (2nd ‘𝑋)⟩)
91	90	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸) → ¬ ⟨0, (2nd ‘𝑋)⟩ = ⟨0, (2nd ‘𝑋)⟩)
92	91	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∈ USGraph → ({⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸 → ¬ ⟨0, (2nd ‘𝑋)⟩ = ⟨0, (2nd ‘𝑋)⟩))
93	53, 92	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸 → ¬ ⟨0, (2nd ‘𝑋)⟩ = ⟨0, (2nd ‘𝑋)⟩))
94	89, 93	mt2i 137	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ¬ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸)
95		df-nel 3037	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ ¬ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∈ 𝐸)
96	94, 95	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸)
97	6, 7, 8, 31	gpg5nbgrvtx13starlem3 48549	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (2nd ‘𝑋) ∈ V) → {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
98	61, 97	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
99		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩})
100		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
101	99, 100	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
102		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → {⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩})
103		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
105		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
106		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨0, (2nd ‘𝑋)⟩, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
107	105, 106	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
108	71, 72, 73, 101, 104, 107	raltp 4649	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨0, (2nd ‘𝑋)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
109	88, 96, 98, 108	syl3anbrc 1345	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸)
110		prcom 4676	. . . . . . . . . . . . . . . . . . . 20 ⊢ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}
111		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
112	110, 111	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
113	70, 112	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸)
114		prcom 4676	. . . . . . . . . . . . . . . . . . . 20 ⊢ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}
115		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} = {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
116	114, 115	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
117	98, 116	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸)
118		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩
119	31	usgredgne 29275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸) → ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ ≠ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
120	119	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸) → ¬ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩)
121	120	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∈ USGraph → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
122	53, 121	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩))
123	118, 122	mt2i 137	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ¬ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸)
124		df-nel 3037	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸 ↔ ¬ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∈ 𝐸)
125	123, 124	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸)
126		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩})
127		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸))
129		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩})
130		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
131	129, 130	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨0, (2nd ‘𝑋)⟩ → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸))
132		preq2 4678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
133		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
134	132, 133	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
135	71, 72, 73, 128, 131, 134	raltp 4649	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} ∉ 𝐸))
136	113, 117, 125, 135	syl3anbrc 1345	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸)
137		preq1 4677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦})
138		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥, 𝑦} = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
139	137, 138	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
140	139	ralbidv 3160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩ → (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
141		preq1 4677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨0, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, 𝑦})
142		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥, 𝑦} = {⟨0, (2nd ‘𝑋)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨0, (2nd ‘𝑋)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
144	143	ralbidv 3160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨0, (2nd ‘𝑋)⟩ → (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
145		preq1 4677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦})
146		neleq1 3042	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥, 𝑦} = {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
147	145, 146	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
148	147	ralbidv 3160	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩ → (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
149	71, 72, 73, 140, 144, 148	raltp 4649	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ (∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ∧ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨0, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ∧ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
150	84, 109, 136, 149	syl3anbrc 1345	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ∀𝑥 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸)
151	6, 7, 8, 30	gpgnbgrvtx1 48551	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
152	3, 151	sylanl1 681	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩})
153	152	raleqdv 3295	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸))
154	152, 153	raleqbidv 3311	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑥 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩}∀𝑦 ∈ {⟨1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)⟩, ⟨0, (2nd ‘𝑋)⟩, ⟨1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸))
155	150, 154	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)
156	46, 155	jca 511	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))
157	156	exp43 436	. . . . . . . . . . . . 13 ⊢ (𝑁 = 5 → (𝐾 ∈ 𝐽 → (𝑋 ∈ 𝑉 → ((1st ‘𝑋) = 1 → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))))
158	157	3imp 1111	. . . . . . . . . . . 12 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((1st ‘𝑋) = 1 → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
159	44, 158	syl5 34	. . . . . . . . . . 11 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨1, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
160	159	adantl 481	. . . . . . . . . 10 ⊢ ((𝑎 = 1 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨1, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
161	42, 160	sylbid 240	. . . . . . . . 9 ⊢ ((𝑎 = 1 ∧ (𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉)) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
162	161	ex 412	. . . . . . . 8 ⊢ (𝑎 = 1 → ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
163	39, 162	jaoi 858	. . . . . . 7 ⊢ ((𝑎 = 0 ∨ 𝑎 = 1) → ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
164	12, 163	syl 17	. . . . . 6 ⊢ (𝑎 ∈ {0, 1} → ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
165	164	impcom 407	. . . . 5 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑎 ∈ {0, 1}) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
166	165	a1d 25	. . . 4 ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) ∧ 𝑎 ∈ {0, 1}) → (𝑏 ∈ (0..^𝑁) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
167	166	expimpd 453	. . 3 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((𝑎 ∈ {0, 1} ∧ 𝑏 ∈ (0..^𝑁)) → (𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))))
168	167	rexlimdvv 3193	. 2 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (∃𝑎 ∈ {0, 1}∃𝑏 ∈ (0..^𝑁)𝑋 = ⟨𝑎, 𝑏⟩ → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)))
169	11, 168	mpd 15	1 ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∉ wnel 3036  ∀wral 3051  ∃wrex 3061  Vcvv 3429  {cpr 4569  {ctp 4571  ⟨cop 4573   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  0cc0 11038  1c1 11039   + caddc 11041   ≤ cle 11180   − cmin 11377   / cdiv 11807  2c2 12236  3c3 12237  4c4 12238  5c5 12239  ℤcz 12524  ℤ_≥cuz 12788  ..^cfzo 13608  ⌈cceil 13750   mod cmo 13828  ♯chash 14292  Vtxcvtx 29065  Edgcedg 29116  USGraphcusgr 29218   NeighbVtx cnbgr 29401   gPetersenGr cgpg 48516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-ico 13304  df-fz 13462  df-fzo 13609  df-fl 13751  df-ceil 13752  df-mod 13829  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-dvds 16222  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-edgf 29058  df-vtx 29067  df-iedg 29068  df-edg 29117  df-upgr 29151  df-umgr 29152  df-usgr 29220  df-nbgr 29402  df-gpg 48517

This theorem is referenced by:  gpg5gricstgr3  48566