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Theorem frgrregord013 30414

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
frgrreggt1.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
frgrregord013	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Proof of Theorem frgrregord013 Dummy variables 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashcl 14395	. . 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 1106	. . . . . 6 ⊢ (¬ ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3) ↔ (¬ (♯‘𝑉) = 0 ∧ ¬ (♯‘𝑉) = 1 ∧ ¬ (♯‘𝑉) = 3))
4		df-ne 2941	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ≠ 0 ↔ ¬ (♯‘𝑉) = 0)
5		hasheq0 14402	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 2985	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 12540	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑉) ∈ ℕ ↔ ((♯‘𝑉) ∈ ℕ₀ ∧ (♯‘𝑉) ≠ 0))
9		df-ne 2941	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
10		eluz2b3 12964	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) ↔ ((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1))
11		hash2prde 14509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		vex 3484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ∈ V)
14		vex 3484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑏 ∈ V
15	14	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ∈ V)
16		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏)
17	13, 15, 16	3jca 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 ≠ 𝑏 → (𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏))
18		frgrreggt1.v	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑉 = (Vtx‘𝐺)
19	18	eqeq1i 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
20	19	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑉 = {𝑎, 𝑏} → (Vtx‘𝐺) = {𝑎, 𝑏})
21		nfrgr2v 30291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏) ∧ (Vtx‘𝐺) = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
22	17, 20, 21	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
23		df-nel 3047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
24	22, 23	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ¬ 𝐺 ∈ FriendGraph )
25	24	pm2.21d 121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
26	25	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
27	26	exlimivv 1932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∃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	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
29	28	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
30	29	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((♯‘𝑉) = 2 → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
31	30	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((♯‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
32	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((♯‘𝑉) = 2 → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
33	32	3imp 1111	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘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	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) = 2 → (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
35		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
36	18, 35	rusgrprop0 29585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
37		eluz2gt1 12962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → 1 < (♯‘𝑉))
38	37	anim1ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → (𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)))
39	18	vdgn0frgrv2 30314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
40	39	impancom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
41	40	ralrimiv 3145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
42		eqeq2 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0))
43	42	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0))
44		r19.26 3111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
45		nne 2944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0)
46	45	bicomi 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
47	46	anbi1i 624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
48		ancom 460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
49		pm3.24 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ¬ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
50	49	bifal 1556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
51	47, 48, 50	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
52	51	ralbii 3093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
53		r19.3rzv 4499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
54		falim 1557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
55	53, 54	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
57	56	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 ⊥ → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
58	52, 57	sylbi 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
59	44, 58	sylbir 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
60	59	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
61	43, 60	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
62	61	com4t 93	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
63	38, 41, 62	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
64	63	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
65	64	com25 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (𝐺 ∈ FriendGraph → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
66	65	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((¬ (♯‘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))))))
67	66	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (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))))))
68	67	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∀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))))))
69	68	3ad2ant3 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((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))))))
70	36, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
71	70	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 0 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
72	71	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
73	18	frrusgrord 30360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))
74	73	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))
75		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → 𝐾 = 2)
76		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
77	75, 76	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
78	77	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
79		2m1e1 12392	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 − 1) = 1
80	79	oveq2i 7442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · (2 − 1)) = (2 · 1)
81		2t1e2 12429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · 1) = 2
82	80, 81	eqtri 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (2 · (2 − 1)) = 2
83	82	oveq1i 7441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
84		2p1e3 12408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2 + 1) = 3
85	83, 84	eqtri 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((2 · (2 − 1)) + 1) = 3
86	78, 85	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
87	86	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐾 = 2 → ((♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) ↔ (♯‘𝑉) = 3))
88		pm2.21 123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (¬ (♯‘𝑉) = 3 → ((♯‘𝑉) = 3 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
89	88	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 3 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
90	89	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((♯‘𝑉) = 3 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
91	87, 90	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝐾 = 2 → ((♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
92	74, 91	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 2 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
93	18	frgrreg 30413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
94	93	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
95	72, 92, 94	mpjaod 861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
96	95	exp32 420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
97	96	com34 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
98	97	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
99	98	exp4c 432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 3 → (¬ (♯‘𝑉) = 2 → ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
100	99	com34 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 3 → ((♯‘𝑉) ∈ (ℤ_≥‘2) → (¬ (♯‘𝑉) = 2 → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
101	100	com25 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → ((♯‘𝑉) ∈ (ℤ_≥‘2) → (¬ (♯‘𝑉) = 2 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
102	101	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → ((♯‘𝑉) ∈ (ℤ_≥‘2) → (¬ (♯‘𝑉) = 2 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
103	102	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → ((♯‘𝑉) ∈ (ℤ_≥‘2) → (¬ (♯‘𝑉) = 2 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
104	103	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 2 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
105	104	3imp 1111	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 2 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
106	105	com3r 87	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ (♯‘𝑉) = 2 → (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
107	34, 106	pm2.61i 182	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
108	107	3exp 1120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
109	10, 108	sylbir 235	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
110	109	ex 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
111	9, 110	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
112	111	com25 99	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑉) ∈ ℕ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
113	8, 112	sylbir 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑉) ∈ ℕ₀ ∧ (♯‘𝑉) ≠ 0) → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
114	113	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ((♯‘𝑉) ≠ 0 → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))))
115	114	impcomd 411	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
116	115	com14 96	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ≠ ∅ → ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → ((♯‘𝑉) ∈ ℕ₀ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
117	7, 116	mpcom 38	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → (𝐺 ∈ FriendGraph → ((♯‘𝑉) ∈ ℕ₀ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
118	117	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ≠ 0 → (𝐺 ∈ FriendGraph → ((♯‘𝑉) ∈ ℕ₀ → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
119	118	com14 96	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℕ₀ → ((♯‘𝑉) ≠ 0 → (𝐺 ∈ FriendGraph → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
120	4, 119	biimtrrid 243	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (¬ (♯‘𝑉) = 0 → (𝐺 ∈ FriendGraph → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
121	120	com24 95	. . . . . . . . . . 11 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 0 → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
122	121	3imp 1111	. . . . . . . . . 10 ⊢ (((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) → (¬ (♯‘𝑉) = 0 → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
123	122	com25 99	. . . . . . . . 9 ⊢ (((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) → (𝐺 RegUSGraph 𝐾 → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (¬ (♯‘𝑉) = 0 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
124	123	imp 406	. . . . . . . 8 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → (¬ (♯‘𝑉) = 0 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
125	124	com14 96	. . . . . . 7 ⊢ (¬ (♯‘𝑉) = 0 → (¬ (♯‘𝑉) = 1 → (¬ (♯‘𝑉) = 3 → ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
126	125	3imp 1111	. . . . . 6 ⊢ ((¬ (♯‘𝑉) = 0 ∧ ¬ (♯‘𝑉) = 1 ∧ ¬ (♯‘𝑉) = 3) → ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
127	3, 126	sylbi 217	. . . . 5 ⊢ (¬ ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3) → ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
128	2, 127	pm2.61i 182	. . . 4 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
129	128	3exp1 1353	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
130	1, 129	mpcom 38	. 2 ⊢ (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
131	130	3imp21 1114	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 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1540 ⊥wfal 1552 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∀wral 3061 Vcvv 3480 ∅c0 4333 {cpr 4628 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 ℕcn 12266 2c2 12321 3c3 12322 ℕ₀cn0 12526 ℕ₀^*cxnn0 12599 ℤ_≥cuz 12878 ♯chash 14369 Vtxcvtx 29013 USGraphcusgr 29166 VtxDegcvtxdg 29483 RegUSGraph crusgr 29574 FriendGraph cfrgr 30277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-xadd 13155 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-reps 14807 df-csh 14827 df-s2 14887 df-s3 14888 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-dvds 16291 df-gcd 16532 df-prm 16709 df-phi 16803 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-ushgr 29076 df-upgr 29099 df-umgr 29100 df-uspgr 29167 df-usgr 29168 df-fusgr 29334 df-nbgr 29350 df-vtxdg 29484 df-rgr 29575 df-rusgr 29576 df-wlks 29617 df-wlkson 29618 df-trls 29710 df-trlson 29711 df-pths 29734 df-spths 29735 df-pthson 29736 df-spthson 29737 df-wwlks 29850 df-wwlksn 29851 df-wwlksnon 29852 df-wspthsn 29853 df-wspthsnon 29854 df-clwwlk 30001 df-clwwlkn 30044 df-clwwlknon 30107 df-conngr 30206 df-frgr 30278

This theorem is referenced by: frgrregord13 30415

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