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Theorem frgrreg 30486

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 462	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ↔ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin))
2		ancom 462	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) ↔ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ))
3	1, 2	anbi12i 635	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) ↔ ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
4	3	biimpi 218	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
5	4	ancomd 463	. . . . . 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 30481	. . . . . 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 3029	. . . . . . . 8 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ↔ ¬ (𝐾 = 0 ∨ 𝐾 = 2))
10		nn0re 12441	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℝ)
11		1re 11139	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
12		lenlt 11219	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
13	10, 11, 12	sylancl 593	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
14	13	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
15		elnnne0 12446	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0))
16		nnle1eq1 12202	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 ↔ 𝐾 = 1))
17	16	biimpd 231	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 → 𝐾 = 1))
18	15, 17	sylbir 237	. . . . . . . . . . . . . . 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 455	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ≤ 1 → 𝐾 = 1)))
21	20	impcom 409	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 → 𝐾 = 1))
22	14, 21	sylbird 262	. . . . . . . . . . 11 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → 𝐾 = 1))
23	6	fveq2i 6834	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘𝑉) = (♯‘(Vtx‘𝐺))
24	23	eqeq1i 2746	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 1 ↔ (♯‘(Vtx‘𝐺)) = 1)
25	24	biimpi 218	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) = 1 → (♯‘(Vtx‘𝐺)) = 1)
26		simpr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
27	26	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 RegUSGraph 𝐾)
28		rusgr1vtx 29679	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)
29	25, 27, 28	syl2an 603	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → 𝐾 = 0)
30	29	orcd 880	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → (𝐾 = 0 ∨ 𝐾 = 2))
31	30	ex 414	. . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
34	6, 33	rusgrprop0 29658	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
35		simp2 1144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝐺 ∈ FriendGraph )
36		hashnncl 14323	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅))
37		df-ne 2937	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
38		nngt1ne1 12201	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) ∈ ℕ → (1 < (♯‘𝑉) ↔ (♯‘𝑉) ≠ 1))
39	38	biimprd 250	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → 1 < (♯‘𝑉)))
40	37, 39	biimtrrid 245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
41	36, 40	biimtrrdi 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉))))
42	41	imp 408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
43	42	impcom 409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < (♯‘𝑉))
44	6	vdgn1frgrv3 30389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
45	35, 43, 44	3imp3i2an 1353	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
46		r19.26 3101	. . . . . . . . . . . . . . . . . . . . . . . . 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 4430	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → ∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
48		neeq1 2998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ↔ 𝐾 ≠ 1))
49	48	biimpd 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 → 𝐾 ≠ 1))
50	49	impcom 409	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ≠ 1)
51		eqneqall 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3138	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
57	46, 56	biimtrrid 245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ≠ ∅ → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
58	57	expd 417	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
68	67	impcom 409	. . . . . . . . . . . . 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 183	. . . . . . . . . . 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 414	. . . . . . . . 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 237	. . . . . . 7 ⊢ (¬ (𝐾 = 0 ∨ 𝐾 = 2) → (¬ 1 < 𝐾 → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
75	74	impcom 409	. . . . . 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 432	. 2 ⊢ (¬ 1 < 𝐾 → (¬ (𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))))
80		simprl 777	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 ∈ FriendGraph )
81		simpl 484	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ∈ Fin)
82	81	ad2antlr 734	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ∈ Fin)
83		simpr 486	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ≠ ∅)
84	83	ad2antlr 734	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ≠ ∅)
85		simpl 484	. . . . . 6 ⊢ ((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < 𝐾)
86	85, 26	anim12ci 621	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾))
87	6	frgrreggt1 30485	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))
88	87	imp 408	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾)) → 𝐾 = 2)
89	80, 82, 84, 86, 88	syl31anc 1382	. . . 4 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 = 2)
90	89	olcd 881	. . 3 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
91	90	exp31 421	. 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 184	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 208 ∧ wa 397 ∨ wo 854 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∃wrex 3065 ∅c0 4264 class class class wbr 5075 ‘cfv 6489 Fincfn 8887 ℝcr 11032 0cc0 11033 1c1 11034 < clt 11174 ≤ cle 11175 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ℕ₀^*cxnn0 12505 ♯chash 14287 Vtxcvtx 29087 USGraphcusgr 29240 VtxDegcvtxdg 29556 RegUSGraph crusgr 29647 FriendGraph cfrgr 30350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-ifp 1070 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-disj 5043 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-rp 12938 df-xadd 13059 df-ico 13299 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-reps 14726 df-csh 14746 df-s2 14805 df-s3 14806 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-dvds 16217 df-gcd 16459 df-prm 16636 df-phi 16731 df-vtx 29089 df-iedg 29090 df-edg 29139 df-uhgr 29149 df-ushgr 29150 df-upgr 29173 df-umgr 29174 df-uspgr 29241 df-usgr 29242 df-fusgr 29408 df-nbgr 29424 df-vtxdg 29557 df-rgr 29648 df-rusgr 29649 df-wlks 29690 df-wlkson 29691 df-trls 29781 df-trlson 29782 df-pths 29804 df-spths 29805 df-pthson 29806 df-spthson 29807 df-wwlks 29920 df-wwlksn 29921 df-wwlksnon 29922 df-wspthsn 29923 df-wspthsnon 29924 df-clwwlk 30074 df-clwwlkn 30117 df-clwwlknon 30180 df-frgr 30351

This theorem is referenced by: frgrregord013 30487

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