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Theorem frgrreg 30414

Description: If a finite nonempty friendship graph is 𝐾-regular, then 𝐾 must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)

**Hypothesis**
Ref	Expression
frgrreggt1.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
frgrreg	⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Proof of Theorem frgrreg Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancom 460	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ↔ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin))
2		ancom 460	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) ↔ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ))
3	1, 2	anbi12i 628	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) ↔ ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
4	3	biimpi 216	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
5	4	ancomd 461	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)))
6		frgrreggt1.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	numclwwlk7lem 30409	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ₀)
8	5, 7	syl 17	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 ∈ ℕ₀)
9		neanior 3034	. . . . . . . 8 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ↔ ¬ (𝐾 = 0 ∨ 𝐾 = 2))
10		nn0re 12537	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℝ)
11		1re 11262	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
12		lenlt 11340	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
13	10, 11, 12	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
15		elnnne0 12542	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0))
16		nnle1eq1 12297	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 ↔ 𝐾 = 1))
17	16	biimpd 229	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 → 𝐾 = 1))
18	15, 17	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0) → (𝐾 ≤ 1 → 𝐾 = 1))
19	18	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0) → (𝐾 ≠ 2 → (𝐾 ≤ 1 → 𝐾 = 1)))
20	19	expimpd 453	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ≤ 1 → 𝐾 = 1)))
21	20	impcom 407	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 → 𝐾 = 1))
22	14, 21	sylbird 260	. . . . . . . . . . 11 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → 𝐾 = 1))
23	6	fveq2i 6908	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘𝑉) = (♯‘(Vtx‘𝐺))
24	23	eqeq1i 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 1 ↔ (♯‘(Vtx‘𝐺)) = 1)
25	24	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) = 1 → (♯‘(Vtx‘𝐺)) = 1)
26		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
27	26	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 RegUSGraph 𝐾)
28		rusgr1vtx 29607	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)
29	25, 27, 28	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → 𝐾 = 0)
30	29	orcd 873	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → (𝐾 = 0 ∨ 𝐾 = 2))
31	30	ex 412	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
32	31	a1d 25	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) = 1 → (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
33		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
34	6, 33	rusgrprop0 29586	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
35		simp2 1137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝐺 ∈ FriendGraph )
36		hashnncl 14406	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅))
37		df-ne 2940	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
38		nngt1ne1 12296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) ∈ ℕ → (1 < (♯‘𝑉) ↔ (♯‘𝑉) ≠ 1))
39	38	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → 1 < (♯‘𝑉)))
40	37, 39	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
41	36, 40	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉))))
42	41	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
43	42	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < (♯‘𝑉))
44	6	vdgn1frgrv3 30317	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
45	35, 43, 44	3imp3i2an 1345	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
46		r19.26 3110	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ↔ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
47		r19.2z 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → ∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
48		neeq1 3002	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ↔ 𝐾 ≠ 1))
49	48	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 → 𝐾 ≠ 1))
50	49	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ≠ 1)
51		eqneqall 2950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐾 = 1 → (𝐾 ≠ 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
52	51	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐾 ≠ 1 → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
53	50, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
54	53	rexlimivw 3150	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
55	47, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
56	55	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
57	46, 56	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ≠ ∅ → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
58	57	expd 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
59	58	com34 91	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2)))))
60	59	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2)))))
61	60	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2)))))
62	45, 61	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2))))
63	62	3exp 1119	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (♯‘𝑉) = 1 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2))))))
64	63	com15 101	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))))
65	64	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))))
66	34, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))))
67	66	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
68	67	impcom 407	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
69	68	com13 88	. . . . . . . . . . . 12 ⊢ (¬ (♯‘𝑉) = 1 → (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
70	32, 69	pm2.61i 182	. . . . . . . . . . 11 ⊢ (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
71	22, 70	syl6 35	. . . . . . . . . 10 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
72	71	ex 412	. . . . . . . . 9 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ∈ ℕ₀ → (¬ 1 < 𝐾 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
73	72	com23 86	. . . . . . . 8 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (¬ 1 < 𝐾 → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
74	9, 73	sylbir 235	. . . . . . 7 ⊢ (¬ (𝐾 = 0 ∨ 𝐾 = 2) → (¬ 1 < 𝐾 → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
75	74	impcom 407	. . . . . 6 ⊢ ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
76	75	com13 88	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 ∈ ℕ₀ → ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (𝐾 = 0 ∨ 𝐾 = 2))))
77	8, 76	mpd 15	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (𝐾 = 0 ∨ 𝐾 = 2)))
78	77	com12 32	. . 3 ⊢ ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
79	78	exp4b 430	. 2 ⊢ (¬ 1 < 𝐾 → (¬ (𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))))
80		simprl 770	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 ∈ FriendGraph )
81		simpl 482	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ∈ Fin)
82	81	ad2antlr 727	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ∈ Fin)
83		simpr 484	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ≠ ∅)
84	83	ad2antlr 727	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ≠ ∅)
85		simpl 482	. . . . . 6 ⊢ ((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < 𝐾)
86	85, 26	anim12ci 614	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾))
87	6	frgrreggt1 30413	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))
88	87	imp 406	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾)) → 𝐾 = 2)
89	80, 82, 84, 86, 88	syl31anc 1374	. . . 4 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 = 2)
90	89	olcd 874	. . 3 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
91	90	exp31 419	. 2 ⊢ (1 < 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))))
92		2a1 28	. 2 ⊢ ((𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))))
93	79, 91, 92	pm2.61ii 183	1 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∅c0 4332 class class class wbr 5142 ‘cfv 6560 Fincfn 8986 ℝcr 11155 0cc0 11156 1c1 11157 < clt 11296 ≤ cle 11297 ℕcn 12267 2c2 12322 ℕ₀cn0 12528 ℕ₀^*cxnn0 12601 ♯chash 14370 Vtxcvtx 29014 USGraphcusgr 29167 VtxDegcvtxdg 29484 RegUSGraph crusgr 29575 FriendGraph cfrgr 30278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

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 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-rp 13036 df-xadd 13156 df-ico 13394 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-hash 14371 df-word 14554 df-lsw 14602 df-concat 14610 df-s1 14635 df-substr 14680 df-pfx 14710 df-reps 14808 df-csh 14828 df-s2 14888 df-s3 14889 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-dvds 16292 df-gcd 16533 df-prm 16710 df-phi 16804 df-vtx 29016 df-iedg 29017 df-edg 29066 df-uhgr 29076 df-ushgr 29077 df-upgr 29100 df-umgr 29101 df-uspgr 29168 df-usgr 29169 df-fusgr 29335 df-nbgr 29351 df-vtxdg 29485 df-rgr 29576 df-rusgr 29577 df-wlks 29618 df-wlkson 29619 df-trls 29711 df-trlson 29712 df-pths 29735 df-spths 29736 df-pthson 29737 df-spthson 29738 df-wwlks 29851 df-wwlksn 29852 df-wwlksnon 29853 df-wspthsn 29854 df-wspthsnon 29855 df-clwwlk 30002 df-clwwlkn 30045 df-clwwlknon 30108 df-frgr 30279

This theorem is referenced by: frgrregord013 30415

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