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Theorem frgraregord013 26645

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.)

**Assertion**
Ref	Expression
frgraregord013	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

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

Step	Hyp	Ref	Expression
1		hashcl 13009	. . . 4 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ₀)
2		ax-1 6	. . . . . 6 ⊢ (((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
3		3ioran 1049	. . . . . . 7 ⊢ (¬ ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) ↔ (¬ (#‘𝑉) = 0 ∧ ¬ (#‘𝑉) = 1 ∧ ¬ (#‘𝑉) = 3))
4		df-ne 2782	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑉) ≠ 0 ↔ ¬ (#‘𝑉) = 0)
5		hasheq0 13015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 11183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑉) ∈ ℕ ↔ ((#‘𝑉) ∈ ℕ₀ ∧ (#‘𝑉) ≠ 0))
9		df-ne 2782	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑉) ≠ 1 ↔ ¬ (#‘𝑉) = 1)
10		eluz2b3 11638	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) ↔ ((#‘𝑉) ∈ ℕ ∧ (#‘𝑉) ≠ 1))
11		hash2prde 13109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		breq1 4586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 FriendGrph 𝐸 ↔ {𝑎, 𝑏} FriendGrph 𝐸))
13	12	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 FriendGrph 𝐸 ↔ {𝑎, 𝑏} FriendGrph 𝐸))
14		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ 𝑎 ∈ V
15		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ 𝑏 ∈ V
16	14, 15	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑎 ∈ V ∧ 𝑏 ∈ V)
17	16	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ V ∧ 𝑏 ∈ V))
18	17	anim1i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑉 = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏) → ((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ 𝑎 ≠ 𝑏))
19	18	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ 𝑎 ≠ 𝑏))
20		frgra2v 26526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑎 ∈ V ∧ 𝑏 ∈ V) ∧ 𝑎 ≠ 𝑏) → ¬ {𝑎, 𝑏} FriendGrph 𝐸)
21	19, 20	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ¬ {𝑎, 𝑏} FriendGrph 𝐸)
22	21	pm2.21d 117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ({𝑎, 𝑏} FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
23	13, 22	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
24	23	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
25	24	exlimivv 1847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
26	11, 25	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) = 2) → (𝑉 ≠ ∅ → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
27	26	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 2 → (𝑉 ≠ ∅ → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
28	27	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
29	28	com14 94	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
30	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
31	30	3imp 1249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
32	31	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑉) = 2 → (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
33		rusgraprop 26456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ₀ ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾))
34		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)))
35		1red 9934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 1 ∈ ℝ)
36		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ 2 ∈ ℝ
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 2 ∈ ℝ)
38		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((#‘𝑉) ∈ ℤ → (#‘𝑉) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → (#‘𝑉) ∈ ℝ)
40		1lt2 11071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ 1 < 2
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 1 < 2)
42		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 2 ≤ (#‘𝑉))
43	35, 37, 39, 41, 42	ltletrd 10076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (((#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 1 < (#‘𝑉))
44	43	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((2 ∈ ℤ ∧ (#‘𝑉) ∈ ℤ ∧ 2 ≤ (#‘𝑉)) → 1 < (#‘𝑉))
45	34, 44	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → 1 < (#‘𝑉))
46	45	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)))
47	46	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸) → (𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)))
48		vdgn0frgrav2 26551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑣 ∈ 𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0))
49	48	impancom 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → (𝑣 ∈ 𝑉 → ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0))
50	49	ralrimiv 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0)
51		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (𝐾 = 0 → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ↔ ((𝑉 VDeg 𝐸)‘𝑣) = 0))
52	51	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0))
53		r19.26 3046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0))
54		nne 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ⊢ (¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ↔ ((𝑉 VDeg 𝐸)‘𝑣) = 0)
55	54	bicomi 213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (((𝑉 VDeg 𝐸)‘𝑣) = 0 ↔ ¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0)
56	55	anbi1i 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ (¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0))
57		ancom 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ (((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ∧ ¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0))
58		pm3.24 922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ ¬ (((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ∧ ¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0)
59	58	bifal 1488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ ((((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 ∧ ¬ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ ⊥)
60	56, 57, 59	3bitri 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ ⊥)
61	60	ralbii 2963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
62		r19.3rzv 4016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
63		falim 1489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ⊢ (⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))
64	62, 63	syl6bir 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
65	64	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ⊥ → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
66	65	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (∀𝑣 ∈ 𝑉 ⊥ → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
67	61, 66	sylbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
68	53, 67	sylbir 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ ((∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
69	68	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
70	52, 69	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
71	70	com4t 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
72	47, 50, 71	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
73	72	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝑉 FriendGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
74	73	com25 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
75	74	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
76	75	com15 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑉 FriendGrph 𝐸 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
77	76	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 FriendGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
78	77	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ₀ ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 FriendGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
79	33, 78	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 FriendGrph 𝐸 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
80	79	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
81	80	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)) → (𝐾 = 0 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
82		frrusgraord 26598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))
83	82	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))
84		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝐾 = 2 → 𝐾 = 2)
85		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
86	84, 85	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
87	86	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
88		2m1e1 11012	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (2 − 1) = 1
89	88	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (2 · (2 − 1)) = (2 · 1)
90		2t1e2 11053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (2 · 1) = 2
91	89, 90	eqtri 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 · (2 − 1)) = 2
92	91	oveq1i 6559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
93		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 + 1) = 3
94	92, 93	eqtri 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((2 · (2 − 1)) + 1) = 3
95	87, 94	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
96	95	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) ↔ (#‘𝑉) = 3))
97		pm2.21 119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (¬ (#‘𝑉) = 3 → ((#‘𝑉) = 3 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
98	97	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) → ((#‘𝑉) = 3 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
99	98	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 3 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
100	99	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((#‘𝑉) = 3 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
101	96, 100	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
102	83, 101	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)) → (𝐾 = 2 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
103		frgrareg 26644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
104	103	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
105	81, 102, 104	mpjaod 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
106	105	exp32 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
107	106	com34 89	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝑉 FriendGrph 𝐸 → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
108	107	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (((¬ (#‘𝑉) = 3 ∧ ¬ (#‘𝑉) = 2) ∧ (#‘𝑉) ∈ (ℤ_≥‘2)) → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
109	108	exp4c 634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 2 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
110	109	com34 89	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (#‘𝑉) = 3 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
111	110	com25 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
112	111	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
113	112	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → ((#‘𝑉) ∈ (ℤ_≥‘2) → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
114	113	com14 94	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
115	114	3imp 1249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 2 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
116	115	com3r 85	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (¬ (#‘𝑉) = 2 → (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
117	32, 116	pm2.61i 175	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
118	117	3exp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑉) ∈ (ℤ_≥‘2) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
119	10, 118	sylbir 224	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝑉) ∈ ℕ ∧ (#‘𝑉) ≠ 1) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
120	119	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) ≠ 1 → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
121	9, 120	syl5bir 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑉) ∈ ℕ → (¬ (#‘𝑉) = 1 → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
122	121	com25 97	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑉) ∈ ℕ → (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
123	8, 122	sylbir 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ (#‘𝑉) ≠ 0) → (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
124	123	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((#‘𝑉) ≠ 0 → (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))))
125	124	com23 84	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))))
126	125	impd 446	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝑉 FriendGrph 𝐸 → (𝑉 ≠ ∅ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
127	126	com14 94	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ≠ ∅ → ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
128	7, 127	mpcom 37	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (#‘𝑉) ≠ 0) → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))))
129	128	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ≠ 0 → (𝑉 FriendGrph 𝐸 → ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
130	129	com14 94	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑉) ∈ ℕ₀ → ((#‘𝑉) ≠ 0 → (𝑉 FriendGrph 𝐸 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
131	4, 130	syl5bir 232	. . . . . . . . . . . . 13 ⊢ ((#‘𝑉) ∈ ℕ₀ → (¬ (#‘𝑉) = 0 → (𝑉 FriendGrph 𝐸 → (𝑉 ∈ Fin → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
132	131	com24 93	. . . . . . . . . . . 12 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))))
133	132	3imp 1249	. . . . . . . . . . 11 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) → (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
134	133	com25 97	. . . . . . . . . 10 ⊢ (((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 0 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))))
135	134	imp 444	. . . . . . . . 9 ⊢ ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → (¬ (#‘𝑉) = 0 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
136	135	com14 94	. . . . . . . 8 ⊢ (¬ (#‘𝑉) = 0 → (¬ (#‘𝑉) = 1 → (¬ (#‘𝑉) = 3 → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
137	136	3imp 1249	. . . . . . 7 ⊢ ((¬ (#‘𝑉) = 0 ∧ ¬ (#‘𝑉) = 1 ∧ ¬ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
138	3, 137	sylbi 206	. . . . . 6 ⊢ (¬ ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3) → ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))
139	2, 138	pm2.61i 175	. . . . 5 ⊢ ((((#‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝑉 FriendGrph 𝐸) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))
140	139	3exp1 1275	. . . 4 ⊢ ((#‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)))))
141	1, 140	mpcom 37	. . 3 ⊢ (𝑉 ∈ Fin → (𝑉 FriendGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
142	141	com12 32	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 ∈ Fin → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))))
143	142	3imp 1249	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ⊥wfal 1480 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∅c0 3874 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 3c3 10948 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 #chash 12979 USGrph cusg 25859 VDeg cvdg 26420 RegUSGrph crusgra 26450 FriendGrph cfrgra 26515

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-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-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-ot 4134 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-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-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-usgra 25862 df-nbgra 25949 df-wlk 26036 df-trail 26037 df-pth 26038 df-spth 26039 df-wlkon 26042 df-spthon 26045 df-wwlk 26207 df-wwlkn 26208 df-clwwlk 26279 df-clwwlkn 26280 df-2wlkonot 26385 df-2spthonot 26387 df-2spthsot 26388 df-vdgr 26421 df-rgra 26451 df-rusgra 26452 df-frgra 26516

This theorem is referenced by: frgraregord13 26646

W3C validator