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Theorem frgrregord013 28759

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 14071	. . 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 1105	. . . . . 6 ⊢ (¬ ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3) ↔ (¬ (♯‘𝑉) = 0 ∧ ¬ (♯‘𝑉) = 1 ∧ ¬ (♯‘𝑉) = 3))
4		df-ne 2944	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ≠ 0 ↔ ¬ (♯‘𝑉) = 0)
5		hasheq0 14078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 2988	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 12247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑉) ∈ ℕ ↔ ((♯‘𝑉) ∈ ℕ₀ ∧ (♯‘𝑉) ≠ 0))
9		df-ne 2944	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
10		eluz2b3 12662	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) ↔ ((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1))
11		hash2prde 14184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ∈ V)
14		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 ≠ 𝑏 → (𝑎 ∈ V ∧ 𝑏 ∈ V ∧ 𝑎 ≠ 𝑏))
18		frgrreggt1.v	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑉 = (Vtx‘𝐺)
19	18	eqeq1i 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
20	19	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑉 = {𝑎, 𝑏} → (Vtx‘𝐺) = {𝑎, 𝑏})
21		nfrgr2v 28636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3050	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
24	22, 23	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
36	18, 35	rusgrprop0 27934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
37		eluz2gt1 12660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → 1 < (♯‘𝑉))
38	37	anim1ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → (𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)))
39	18	vdgn0frgrv2 28659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
40	39	impancom 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → (𝑣 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
41	40	ralrimiv 3102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
42		eqeq2 2750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0))
43	42	ralbidv 3112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0))
44		r19.26 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ⊢ (¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0)
46	45	bicomi 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ ((¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0))
49		pm3.24 403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ ¬ (((VtxDeg‘𝐺)‘𝑣) ≠ 0 ∧ ¬ ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
50	49	bifal 1555	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
53		r19.3rzv 4429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
54		falim 1556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
55	53, 54	syl6bir 253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ⊥ → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
56	55	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))
59	44, 58	sylbir 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 28705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))
74	73	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))
75		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → 𝐾 = 2)
76		oveq1 7282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
77	75, 76	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
78	77	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
79		2m1e1 12099	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 − 1) = 1
80	79	oveq2i 7286	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · (2 − 1)) = (2 · 1)
81		2t1e2 12136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · 1) = 2
82	80, 81	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (2 · (2 − 1)) = 2
83	82	oveq1i 7285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
84		2p1e3 12115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2 + 1) = 3
85	83, 84	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((2 · (2 − 1)) + 1) = 3
86	78, 85	eqtrdi 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
87	86	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 28758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
94	93	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
95	72, 92, 94	mpjaod 857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
109	10, 108	sylbir 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1) → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))))))
110	109	ex 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → (𝐺 ∈ FriendGraph → (𝑉 ≠ ∅ → (𝑉 ∈ Fin → (¬ (♯‘𝑉) = 3 → (𝐺 RegUSGraph 𝐾 → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))))))
111	9, 110	syl5bir 242	. . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . 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	syl5bir 242	. . . . . . . . . . . 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 1110	. . . . . . . . . 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 407	. . . . . . . 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 1110	. . . . . 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 216	. . . . 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 1351	. . 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 1113	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 396 ∨ wo 844 ∨ w3o 1085 ∧ w3a 1086 = wceq 1539 ⊥wfal 1551 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∉ wnel 3049 ∀wral 3064 Vcvv 3432 ∅c0 4256 {cpr 4563 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 − cmin 11205 ℕcn 11973 2c2 12028 3c3 12029 ℕ₀cn0 12233 ℕ₀^*cxnn0 12305 ℤ_≥cuz 12582 ♯chash 14044 Vtxcvtx 27366 USGraphcusgr 27519 VtxDegcvtxdg 27832 RegUSGraph crusgr 27923 FriendGraph cfrgr 28622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-ac2 10219 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-ac 9872 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-xadd 12849 df-ico 13085 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-reps 14482 df-csh 14502 df-s2 14561 df-s3 14562 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-gcd 16202 df-prm 16377 df-phi 16467 df-vtx 27368 df-iedg 27369 df-edg 27418 df-uhgr 27428 df-ushgr 27429 df-upgr 27452 df-umgr 27453 df-uspgr 27520 df-usgr 27521 df-fusgr 27684 df-nbgr 27700 df-vtxdg 27833 df-rgr 27924 df-rusgr 27925 df-wlks 27966 df-wlkson 27967 df-trls 28060 df-trlson 28061 df-pths 28084 df-spths 28085 df-pthson 28086 df-spthson 28087 df-wwlks 28195 df-wwlksn 28196 df-wwlksnon 28197 df-wspthsn 28198 df-wspthsnon 28199 df-clwwlk 28346 df-clwwlkn 28389 df-clwwlknon 28452 df-conngr 28551 df-frgr 28623

This theorem is referenced by: frgrregord13 28760

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