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Theorem frgrregord013 30757

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 14399	. . 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 1122	. . . . . 6 ⊢ (¬ ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3) ↔ (¬ (♯‘𝑉) = 0 ∧ ¬ (♯‘𝑉) = 1 ∧ ¬ (♯‘𝑉) = 3))
4		df-ne 2958	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ≠ 0 ↔ ¬ (♯‘𝑉) = 0)
5		hasheq0 14406	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 3001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 12524	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑉) ∈ ℕ ↔ ((♯‘𝑉) ∈ ℕ₀ ∧ (♯‘𝑉) ≠ 0))
9		df-ne 2958	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
10		eluz2b3 12952	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) ↔ ((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1))
11		hash2prde 14514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		vex 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ∈ V)
14		vex 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑏 ∈ V
15	14	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑏 ∈ V)
16		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏)
17	13, 15, 16	3jca 1145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 ≠ 𝑏 → (𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏))
18		frgrreggt1.v	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑉 = (Vtx‘𝐺)
19	18	eqeq1i 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
20	19	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑉 = {𝑎, 𝑏} → (Vtx‘𝐺) = {𝑎, 𝑏})
21		nfrgr2v 30634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏) ∧ (Vtx‘𝐺) = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
22	17, 20, 21	syl2an 607	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝐺 ∉ FriendGraph )
23		df-nel 3064	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
24	22, 23	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ¬ 𝐺 ∈ FriendGraph )
25	24	pm2.21d 122	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
26	25	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
27	26	exlimivv 1961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
29	28	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 → (𝑉 ≠ ∅ → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
30	29	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((♯‘𝑉) = 2 → (𝐺 ∈ FriendGraph → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
31	30	com14 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) = 2 → (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))
35		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
36	18, 35	rusgrprop0 29928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
37		eluz2gt1 12950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → 1 < (♯‘𝑉))
38	37	anim1ci 627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → (𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)))
39	18	vdgn0frgrv2 30657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
40	39	impancom 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
41	40	ralrimiv 3155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
42		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0))
43	42	ralbidv 3187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0))
44		r19.26 3124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0)
46	45	bicomi 227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 0 ↔ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
47	46	anbi1i 635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
48		ancom 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
49		pm3.24 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ¬ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
50	49	bifal 1585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
51	47, 48, 50	3bitri 300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ⊥)
52	51	ralbii 3110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
53		r19.3rzv 4463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
54		falim 1586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
55	53, 54	biimtrrdi 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
56	55	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
57	56	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 ⊥ → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
58	52, 57	sylbi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
59	44, 58	sylbir 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 94	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 30703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))
74	73	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))
75		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → 𝐾 = 2)
76		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
77	75, 76	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
78	77	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
79		2m1e1 12371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 − 1) = 1
80	79	oveq2i 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · (2 − 1)) = (2 · 1)
81		2t1e2 12409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · 1) = 2
82	80, 81	eqtri 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (2 · (2 − 1)) = 2
83	82	oveq1i 7422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
84		2p1e3 12388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2 + 1) = 3
85	83, 84	eqtri 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((2 · (2 − 1)) + 1) = 3
86	78, 85	eqtrdi 2813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
87	86	eqeq2d 2773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝐾 = 2 → ((♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1) ↔ (♯‘𝑉) = 3))
88		pm2.21 124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (¬ (♯‘𝑉) = 3 → ((♯‘𝑉) = 3 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
89	88	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 3 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
90	89	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((♯‘𝑉) = 3 → (((¬ (♯‘𝑉) = 3 ∧ ¬ (♯‘𝑉) = 2) ∧ (♯‘𝑉) ∈ (ℤ_≥‘2)) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
91	87, 90	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 30756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
94	93	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
95	72, 92, 94	mpjaod 873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 97	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 88	. . . . . . . . . . . . . . . . . . . . . . . . . 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 184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅) → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
108	107	3exp 1136	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
109	10, 108	sylbir 238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
110	109	ex 417	. . . . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
112	111	com25 100	. . . . . . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . 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 97	. . . . . . . . . . . . . . . 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 39	. . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . 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 97	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 96	. . . . . . . . . . 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 1127	. . . . . . . . . 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 100	. . . . . . . . 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 411	. . . . . . . 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 97	. . . . . . 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 1127	. . . . . 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 220	. . . . 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 184	. . . 4 ⊢ ((((♯‘𝑉) ∈ ℕ₀ ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
129	128	3exp1 1370	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ₀ → (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))
130	1, 129	mpcom 39	. 2 ⊢ (𝑉 ∈ Fin → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))
131	130	3imp21 1130	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 860 ∨ w3o 1101 ∧ w3a 1102 = wceq 1569 ⊥wfal 1581 ∃wex 1808 ∈ wcel 2142 ≠ wne 2957 ∉ wnel 3063 ∀wral 3078 Vcvv 3454 ∅c0 4285 {cpr 4590 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 Fincfn 8941 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11249 − cmin 11447 ℕcn 12239 2c2 12301 3c3 12302 ℕ₀cn0 12510 ℕ₀^*cxnn0 12583 ℤ_≥cuz 12868 ♯chash 14373 Vtxcvtx 29357 USGraphcusgr 29510 VtxDegcvtxdg 29826 RegUSGraph crusgr 29917 FriendGraph cfrgr 30620

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1078 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-xnn0 12584 df-z 12598 df-uz 12869 df-rp 13023 df-xadd 13144 df-ico 13384 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-hash 14374 df-word 14558 df-lsw 14607 df-concat 14615 df-s1 14641 df-substr 14686 df-pfx 14716 df-reps 14813 df-csh 14833 df-s2 14892 df-s3 14893 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-dvds 16317 df-gcd 16559 df-prm 16736 df-phi 16831 df-vtx 29359 df-iedg 29360 df-edg 29409 df-uhgr 29419 df-ushgr 29420 df-upgr 29443 df-umgr 29444 df-uspgr 29511 df-usgr 29512 df-fusgr 29678 df-nbgr 29694 df-vtxdg 29827 df-rgr 29918 df-rusgr 29919 df-wlks 29960 df-wlkson 29961 df-trls 30051 df-trlson 30052 df-pths 30074 df-spths 30075 df-pthson 30076 df-spthson 30077 df-wwlks 30190 df-wwlksn 30191 df-wwlksnon 30192 df-wspthsn 30193 df-wspthsnon 30194 df-clwwlk 30344 df-clwwlkn 30387 df-clwwlknon 30450 df-conngr 30549 df-frgr 30621

This theorem is used by: frgrregord13 30758

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