av-frgraregord013 - Mathbox for Alexander van der Vekens

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Theorem av-frgraregord013 41549

Description: If a finite friendship graph is 𝐾-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

**Hypothesis**
Ref	Expression
av-frgrareggt1.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
av-frgraregord013	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Proof of Theorem av-frgraregord013 Dummy variables 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashcl 13009	. . 3 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ₀)
2		ax-1 6	. . . . 5 ⊢ (((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
3		3ioran 1049	. . . . . 6 ⊢ (¬ ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) ↔ (¬ (#‘𝑉) = 0 ∧ ¬ (#‘𝑉) = 1 ∧ ¬ (#‘𝑉) = 3))
4		df-ne 2782	. . . . . . . . . . . . 13 ⊢ ((#‘𝑉) ≠ 0 ↔ ¬ (#‘𝑉) = 0)
5		hasheq0 13015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 500	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 11183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑉) ∈ ℕ ↔ ((#‘𝑉) ∈ ℕ₀ ∧ (#‘𝑉) ≠ 0))
9		df-ne 2782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑉) ≠ 1 ↔ ¬ (#‘𝑉) = 1)
10		eluz2b3 11638	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) ↔ ((#‘𝑉) ∈ ℕ ∧ (#‘𝑉) ≠ 1))
11		hash2prde 13109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑎 ∈ V
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ∈ V)
14		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑏 ∈ V
15	14	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ∈ V)
16		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏)
17	13, 15, 16	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → (𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏))
18		av-frgrareggt1.v	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑉 = (Vtx‘𝐺)
19	18	eqeq1i 2615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
20	19	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} → (Vtx‘𝐺) = {𝑎, 𝑏})
21		nfrgr2v 41442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏) ∧ (Vtx‘𝐺) = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
22	17, 20, 21	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
23		df-nel 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
24	22, 23	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ¬ 𝐺 ∈ FriendGraph )
25	24	pm2.21d 117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
26	25	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
27	26	exlimivv 1847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
28	11, 27	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) = 2) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
29	28	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 2 → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
30	29	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
31	30	com14 94	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
33	32	3imp 1249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
34	33	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑉) = 2 → (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
35		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
36	18, 35	rusgrprop0 40767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
37		eluz2gt1 11636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → 1 < (#‘𝑉))
38	37	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝐺 ∈ FriendGraph ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)))
39	38	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → (𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)))
40	18	vdgn0frgrv2 41465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
41	40	impancom 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
42	41	ralrimiv 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
43		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝐾 = 0 → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0))
44	43	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0))
45		r19.26 3046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
46		nne 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0)
47	46	bicomi 213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
48	47	anbi1i 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
49		ancom 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
50		pm3.24 922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ¬ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
51	50	bifal 1488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
52	48, 49, 51	3bitri 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
53	52	ralbii 2963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
54		r19.3rzv 4016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
55		falim 1489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))
56	54, 55	syl6bir 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
58	57	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (∀𝑣 ∈ 𝑉 ⊥ → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
59	53, 58	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
60	45, 59	sylbir 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
61	60	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
62	44, 61	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
63	62	com4t 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
64	39, 42, 63	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
65	64	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
66	65	com25 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (𝐺 ∈ FriendGraph → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
67	66	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (𝐺 ∈ FriendGraph → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
68	67	com15 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
69	68	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
70	69	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
71	36, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
72	71	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
73	72	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
74	18	frrusgrord 41504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))
75	74	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))
76		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝐾 = 2 → 𝐾 = 2)
77		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
78	76, 77	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
79	78	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
80		2m1e1 11012	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (2 − 1) = 1
81	80	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 · (2 − 1)) = (2 · 1)
82		2t1e2 11053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 · 1) = 2
83	81, 82	eqtri 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · (2 − 1)) = 2
84	83	oveq1i 6559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
85		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (2 + 1) = 3
86	84, 85	eqtri 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((2 · (2 − 1)) + 1) = 3
87	79, 86	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
88	87	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) ↔ (#‘𝑉) = 3))
89		pm2.21 119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (¬ (#‘𝑉) = 3 → ((#‘𝑉) = 3 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
90	89	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 3 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
91	90	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((#‘𝑉) = 3 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
92	88, 91	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐾 = 2 → ((#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
93	75, 92	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 2 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
94	18	av-frgrareg 41548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
95	94	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
96	73, 93, 95	mpjaod 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
97	96	exp32 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
98	97	com34 89	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
99	98	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
100	99	exp4c 634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 2 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
101	100	com34 89	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (#‘𝑉) = 3 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
102	101	com25 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
103	102	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
104	103	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
105	104	com14 94	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
106	105	3imp 1249	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
107	106	com3r 85	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ (#‘𝑉) = 2 → (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
108	34, 107	pm2.61i 175	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
109	108	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
110	10, 109	sylbir 224	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((#‘𝑉) ∈ ℕ ∧ (#‘𝑉) ≠ 1) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
111	110	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) ≠ 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
112	9, 111	syl5bir 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑉) ∈ ℕ → (¬ (#‘𝑉) = 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
113	112	com25 97	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑉) ∈ ℕ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
114	8, 113	sylbir 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ (#‘𝑉) ≠ 0) → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
115	114	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((#‘𝑉) ≠ 0 → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))))
116	115	com23 84	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))))
117	116	impd 446	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
118	117	com14 94	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ≠ ∅ → ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
119	7, 118	mpcom 37	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
120	119	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 → (𝐺 ∈ FriendGraph → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
121	120	com14 94	. . . . . . . . . . . . 13 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((#‘𝑉) ≠ 0 → (𝐺 ∈ FriendGraph → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
122	4, 121	syl5bir 232	. . . . . . . . . . . 12 ⊢ ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 0 → (𝐺 ∈ FriendGraph → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
123	122	com24 93	. . . . . . . . . . 11 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
124	123	3imp 1249	. . . . . . . . . 10 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) → (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
125	124	com25 97	. . . . . . . . 9 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) → (𝐺 RegUSGraph 𝐾 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 0 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
126	125	imp 444	. . . . . . . 8 ⊢ ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 0 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
127	126	com14 94	. . . . . . 7 ⊢ (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
128	127	3imp 1249	. . . . . 6 ⊢ ((¬ (#‘𝑉) = 0 ∧ ¬ (#‘𝑉) = 1 ∧ ¬ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
129	3, 128	sylbi 206	. . . . 5 ⊢ (¬ ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
130	2, 129	pm2.61i 175	. . . 4 ⊢ ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))
131	130	3exp1 1275	. . 3 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
132	1, 131	mpcom 37	. 2 ⊢ (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
133	132	3imp21 1269	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ⊥wfal 1480 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 Vcvv 3173 ∅c0 3874 {cpr 4127 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 ℕcn 10897 2c2 10947 3c3 10948 ℕ₀cn0 11169 ℕ₀^*cxnn0 11240 ℤ_≥cuz 11563 #chash 12979 Vtxcvtx 25673 USGraph cusgr 40379 VtxDegcvtxdg 40681 RegUSGraph crusgr 40756 FriendGraph cfrgr 41428

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-inf2 8421 ax-ac2 9168 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-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-disj 4554 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-se 4998 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-isom 5813 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-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-ac 8822 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-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-reps 13161 df-csh 13386 df-s2 13444 df-s3 13445 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-fusgr 40536 df-nbgr 40554 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 df-1wlks 40800 df-wlks 40801 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-pthson 40925 df-spthson 40926 df-wwlks 41033 df-wwlksn 41034 df-wwlksnon 41035 df-wspthsn 41036 df-wspthsnon 41037 df-clwwlks 41185 df-clwwlksn 41186 df-conngr 41354 df-frgr 41429

This theorem is referenced by: av-frgraregord13 41550

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