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Theorem frgrregord013 30423

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 14391	. . 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 2938	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ≠ 0 ↔ ¬ (♯‘𝑉) = 0)
5		hasheq0 14398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
6	5	necon3bid 2982	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
7	6	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) ≠ 0) → 𝑉 ≠ ∅)
8		elnnne0 12537	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑉) ∈ ℕ ↔ ((♯‘𝑉) ∈ ℕ₀ ∧ (♯‘𝑉) ≠ 0))
9		df-ne 2938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
10		eluz2b3 12961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) ↔ ((♯‘𝑉) ∈ ℕ ∧ (♯‘𝑉) ≠ 1))
11		hash2prde 14505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
12		vex 3481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ 𝑎 ∈ V
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 ≠ 𝑏 → 𝑎 ∈ V)
14		vex 3481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
20	19	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑉 = {𝑎, 𝑏} → (Vtx‘𝐺) = {𝑎, 𝑏})
21		nfrgr2v 30300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3044	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
36	18, 35	rusgrprop0 29599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
37		eluz2gt1 12959	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((♯‘𝑉) ∈ (ℤ_≥‘2) → 1 < (♯‘𝑉))
38	37	anim1ci 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (((♯‘𝑉) ∈ (ℤ_≥‘2) ∧ 𝐺 ∈ FriendGraph ) → (𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)))
39	18	vdgn0frgrv2 30323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 0)
42		eqeq2 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ⊢ (𝐾 = 0 → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ((VtxDeg‘𝐺)‘𝑣) = 0))
43	42	ralbidv 3175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝐾 = 0 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0))
44		r19.26 3108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1552	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 0 ∧ ((VtxDeg‘𝐺)‘𝑣) ≠ 0) ↔ ∀𝑣 ∈ 𝑉 ⊥)
53		r19.3rzv 4504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ⊢ (𝑉 ≠ ∅ → (⊥ ↔ ∀𝑣 ∈ 𝑉 ⊥))
54		falim 1553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 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 30369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝐾 = 2 → (𝐾 − 1) = (2 − 1))
77	75, 76	oveq12d 7448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝐾 = 2 → (𝐾 · (𝐾 − 1)) = (2 · (2 − 1)))
78	77	oveq1d 7445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = ((2 · (2 − 1)) + 1))
79		2m1e1 12389	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ (2 − 1) = 1
80	79	oveq2i 7441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · (2 − 1)) = (2 · 1)
81		2t1e2 12426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (2 · 1) = 2
82	80, 81	eqtri 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (2 · (2 − 1)) = 2
83	82	oveq1i 7440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((2 · (2 − 1)) + 1) = (2 + 1)
84		2p1e3 12405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2 + 1) = 3
85	83, 84	eqtri 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((2 · (2 − 1)) + 1) = 3
86	78, 85	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝐾 = 2 → ((𝐾 · (𝐾 − 1)) + 1) = 3)
87	86	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 30422	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 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 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 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 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 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 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 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1536 ⊥wfal 1548 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∉ wnel 3043 ∀wral 3058 Vcvv 3477 ∅c0 4338 {cpr 4632 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 ℕcn 12263 2c2 12318 3c3 12319 ℕ₀cn0 12523 ℕ₀^*cxnn0 12596 ℤ_≥cuz 12875 ♯chash 14365 Vtxcvtx 29027 USGraphcusgr 29180 VtxDegcvtxdg 29497 RegUSGraph crusgr 29588 FriendGraph cfrgr 30286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-ec 8745 df-qs 8749 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-rp 13032 df-xadd 13152 df-ico 13389 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-word 14549 df-lsw 14597 df-concat 14605 df-s1 14630 df-substr 14675 df-pfx 14705 df-reps 14803 df-csh 14823 df-s2 14883 df-s3 14884 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-dvds 16287 df-gcd 16528 df-prm 16705 df-phi 16799 df-vtx 29029 df-iedg 29030 df-edg 29079 df-uhgr 29089 df-ushgr 29090 df-upgr 29113 df-umgr 29114 df-uspgr 29181 df-usgr 29182 df-fusgr 29348 df-nbgr 29364 df-vtxdg 29498 df-rgr 29589 df-rusgr 29590 df-wlks 29631 df-wlkson 29632 df-trls 29724 df-trlson 29725 df-pths 29748 df-spths 29749 df-pthson 29750 df-spthson 29751 df-wwlks 29859 df-wwlksn 29860 df-wwlksnon 29861 df-wspthsn 29862 df-wspthsnon 29863 df-clwwlk 30010 df-clwwlkn 30053 df-clwwlknon 30116 df-conngr 30215 df-frgr 30287

This theorem is referenced by: frgrregord13 30424

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