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Theorem frgrreg 30715

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 465	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ↔ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin))
2		ancom 465	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) ↔ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ))
3	1, 2	anbi12i 639	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) ↔ ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
4	3	biimpi 219	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
5	4	ancomd 466	. . . . . 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 30710	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ₀)
8	5, 7	syl 18	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 ∈ ℕ₀)
9		neanior 3058	. . . . . . . 8 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ↔ ¬ (𝐾 = 0 ∨ 𝐾 = 2))
10		nn0re 12516	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℝ)
11		1re 11211	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
12		lenlt 11291	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
13	10, 11, 12	sylancl 597	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
14	13	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
15		elnnne0 12521	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0))
16		nnle1eq1 12269	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 ↔ 𝐾 = 1))
17	16	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 → 𝐾 = 1))
18	15, 17	sylbir 238	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0) → (𝐾 ≤ 1 → 𝐾 = 1))
19	18	a1d 26	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0) → (𝐾 ≠ 2 → (𝐾 ≤ 1 → 𝐾 = 1)))
20	19	expimpd 458	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ≤ 1 → 𝐾 = 1)))
21	20	impcom 412	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 → 𝐾 = 1))
22	14, 21	sylbird 263	. . . . . . . . . . 11 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → 𝐾 = 1))
23	6	fveq2i 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘𝑉) = (♯‘(Vtx‘𝐺))
24	23	eqeq1i 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 1 ↔ (♯‘(Vtx‘𝐺)) = 1)
25	24	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) = 1 → (♯‘(Vtx‘𝐺)) = 1)
26		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
27	26	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 RegUSGraph 𝐾)
28		rusgr1vtx 29908	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)
29	25, 27, 28	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → 𝐾 = 0)
30	29	orcd 886	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → (𝐾 = 0 ∨ 𝐾 = 2))
31	30	ex 417	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
32	31	a1d 26	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) = 1 → (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
33		eqid 2770	. . . . . . . . . . . . . . . . 17 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
34	6, 33	rusgrprop0 29887	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
35		simp2 1153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝐺 ∈ FriendGraph )
36		hashnncl 14405	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅))
37		df-ne 2966	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
38		nngt1ne1 12268	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) ∈ ℕ → (1 < (♯‘𝑉) ↔ (♯‘𝑉) ≠ 1))
39	38	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → 1 < (♯‘𝑉)))
40	37, 39	biimtrrid 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
41	36, 40	biimtrrdi 257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉))))
42	41	imp 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
43	42	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < (♯‘𝑉))
44	6	vdgn1frgrv3 30618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
45	35, 43, 44	3imp3i2an 1362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
46		r19.26 3132	. . . . . . . . . . . . . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → ∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
48		neeq1 3027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ↔ 𝐾 ≠ 1))
49	48	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 → 𝐾 ≠ 1))
50	49	impcom 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ≠ 1)
51		eqneqall 2976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐾 = 1 → (𝐾 ≠ 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
52	51	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐾 ≠ 1 → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
53	50, 52	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
54	53	rexlimivw 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
55	47, 54	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))
56	55	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
57	46, 56	biimtrrid 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ≠ ∅ → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
58	57	expd 420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
59	58	com34 92	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2)))))
60	59	adantl 486	. . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2))))
63	62	3exp 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (♯‘𝑉) = 1 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 2))))))
64	63	com15 102	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))))
65	64	3ad2ant3 1151	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ FriendGraph → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))))
67	66	impcom 412	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
68	67	impcom 412	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
69	68	com13 89	. . . . . . . . . . . 12 ⊢ (¬ (♯‘𝑉) = 1 → (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
70	32, 69	pm2.61i 184	. . . . . . . . . . 11 ⊢ (𝐾 = 1 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
71	22, 70	syl6 36	. . . . . . . . . 10 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
72	71	ex 417	. . . . . . . . 9 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ∈ ℕ₀ → (¬ 1 < 𝐾 → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
73	72	com23 87	. . . . . . . 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 238	. . . . . . 7 ⊢ (¬ (𝐾 = 0 ∨ 𝐾 = 2) → (¬ 1 < 𝐾 → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
75	74	impcom 412	. . . . . 6 ⊢ ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))))
76	75	com13 89	. . . . 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 16	. . . 4 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (𝐾 = 0 ∨ 𝐾 = 2)))
78	77	com12 33	. . 3 ⊢ ((¬ 1 < 𝐾 ∧ ¬ (𝐾 = 0 ∨ 𝐾 = 2)) → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))
79	78	exp4b 435	. 2 ⊢ (¬ 1 < 𝐾 → (¬ (𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))))
80		simprl 782	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 ∈ FriendGraph )
81		simpl 487	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ∈ Fin)
82	81	ad2antlr 739	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ∈ Fin)
83		simpr 489	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ≠ ∅)
84	83	ad2antlr 739	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ≠ ∅)
85		simpl 487	. . . . . 6 ⊢ ((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < 𝐾)
86	85, 26	anim12ci 625	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾))
87	6	frgrreggt1 30714	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))
88	87	imp 411	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾)) → 𝐾 = 2)
89	80, 82, 84, 86, 88	syl31anc 1398	. . . 4 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 = 2)
90	89	olcd 887	. . 3 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
91	90	exp31 424	. 2 ⊢ (1 < 𝐾 → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))))
92		2a1 29	. 2 ⊢ ((𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))))
93	79, 91, 92	pm2.61ii 185	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1568 ∈ wcel 2150 ≠ wne 2965 ∀wral 3086 ∃wrex 3096 ∅c0 4294 class class class wbr 5114 ‘cfv 6540 Fincfn 8946 ℝcr 11102 0cc0 11103 1c1 11104 < clt 11246 ≤ cle 11247 ℕcn 12236 2c2 12298 ℕ₀cn0 12507 ℕ₀^*cxnn0 12580 ♯chash 14369 Vtxcvtx 29316 USGraphcusgr 29469 VtxDegcvtxdg 29785 RegUSGraph crusgr 29876 FriendGraph cfrgr 30579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-inf2 9613 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1077 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-disj 5082 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-er 8697 df-ec 8699 df-qs 8703 df-map 8829 df-pm 8830 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-sup 9405 df-inf 9406 df-oi 9475 df-dju 9890 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-n0 12508 df-xnn0 12581 df-z 12595 df-uz 12866 df-rp 13020 df-xadd 13141 df-ico 13381 df-fz 13539 df-fzo 13686 df-fl 13828 df-mod 13906 df-seq 14041 df-exp 14101 df-hash 14370 df-word 14554 df-lsw 14603 df-concat 14611 df-s1 14637 df-substr 14682 df-pfx 14712 df-reps 14809 df-csh 14829 df-s2 14888 df-s3 14889 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-clim 15542 df-sum 15741 df-dvds 16314 df-gcd 16556 df-prm 16733 df-phi 16828 df-vtx 29318 df-iedg 29319 df-edg 29368 df-uhgr 29378 df-ushgr 29379 df-upgr 29402 df-umgr 29403 df-uspgr 29470 df-usgr 29471 df-fusgr 29637 df-nbgr 29653 df-vtxdg 29786 df-rgr 29877 df-rusgr 29878 df-wlks 29919 df-wlkson 29920 df-trls 30010 df-trlson 30011 df-pths 30033 df-spths 30034 df-pthson 30035 df-spthson 30036 df-wwlks 30149 df-wwlksn 30150 df-wwlksnon 30151 df-wspthsn 30152 df-wspthsnon 30153 df-clwwlk 30303 df-clwwlkn 30346 df-clwwlknon 30409 df-frgr 30580

This theorem is referenced by: frgrregord013 30716

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