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Theorem frgrreg 30276

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 459	. . . . . . . . 9 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ↔ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin))
2		ancom 459	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) ↔ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ))
3	1, 2	anbi12i 626	. . . . . . . 8 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) ↔ ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
4	3	biimpi 215	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → ((𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph )))
5	4	ancomd 460	. . . . . 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 30271	. . . . . 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 3024	. . . . . . . 8 ⊢ ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ↔ ¬ (𝐾 = 0 ∨ 𝐾 = 2))
10		nn0re 12514	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℝ)
11		1re 11246	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
12		lenlt 11324	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
13	10, 11, 12	sylancl 584	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
14	13	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 ↔ ¬ 1 < 𝐾))
15		elnnne0 12519	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 ∈ ℕ₀ ∧ 𝐾 ≠ 0))
16		nnle1eq1 12275	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 ↔ 𝐾 = 1))
17	16	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ → (𝐾 ≤ 1 → 𝐾 = 1))
18	15, 17	sylbir 234	. . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ₀ → ((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) → (𝐾 ≤ 1 → 𝐾 = 1)))
21	20	impcom 406	. . . . . . . . . . . 12 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (𝐾 ≤ 1 → 𝐾 = 1))
22	14, 21	sylbird 259	. . . . . . . . . . 11 ⊢ (((𝐾 ≠ 0 ∧ 𝐾 ≠ 2) ∧ 𝐾 ∈ ℕ₀) → (¬ 1 < 𝐾 → 𝐾 = 1))
23	6	fveq2i 6899	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘𝑉) = (♯‘(Vtx‘𝐺))
24	23	eqeq1i 2730	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 1 ↔ (♯‘(Vtx‘𝐺)) = 1)
25	24	biimpi 215	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑉) = 1 → (♯‘(Vtx‘𝐺)) = 1)
26		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
27	26	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 RegUSGraph 𝐾)
28		rusgr1vtx 29474	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)
29	25, 27, 28	syl2an 594	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → 𝐾 = 0)
30	29	orcd 871	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑉) = 1 ∧ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾))) → (𝐾 = 0 ∨ 𝐾 = 2))
31	30	ex 411	. . . . . . . . . . . . 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 2725	. . . . . . . . . . . . . . . . 17 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
34	6, 33	rusgrprop0 29453	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
35		simp2 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 𝐺 ∈ FriendGraph )
36		hashnncl 14361	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅))
37		df-ne 2930	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ≠ 1 ↔ ¬ (♯‘𝑉) = 1)
38		nngt1ne1 12274	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑉) ∈ ℕ → (1 < (♯‘𝑉) ↔ (♯‘𝑉) ≠ 1))
39	38	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑉) ∈ ℕ → ((♯‘𝑉) ≠ 1 → 1 < (♯‘𝑉)))
40	37, 39	biimtrrid 242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑉) ∈ ℕ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
41	36, 40	biimtrrdi 253	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉))))
42	41	imp 405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (¬ (♯‘𝑉) = 1 → 1 < (♯‘𝑉)))
43	42	impcom 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ (♯‘𝑉) = 1 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < (♯‘𝑉))
44	6	vdgn1frgrv3 30179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
45	35, 43, 44	3imp3i2an 1342	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ (♯‘𝑉) = 1 ∧ 𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1)
46		r19.26 3100	. . . . . . . . . . . . . . . . . . . . . . . . 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 4496	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) → ∃𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾))
48		neeq1 2992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ↔ 𝐾 ≠ 1))
49	48	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) ≠ 1 → 𝐾 ≠ 1))
50	49	impcom 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝐾 ≠ 1)
51		eqneqall 2940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3140	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
57	46, 56	biimtrrid 242	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑉 ≠ ∅ → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) ≠ 1 ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 = 1 → (𝐾 = 0 ∨ 𝐾 = 2))))
58	57	expd 414	. . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐾 = 1 → (¬ (♯‘𝑉) = 1 → (𝐾 = 0 ∨ 𝐾 = 2)))))
68	67	impcom 406	. . . . . . . . . . . . 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 411	. . . . . . . . 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 234	. . . . . . 7 ⊢ (¬ (𝐾 = 0 ∨ 𝐾 = 2) → (¬ 1 < 𝐾 → (𝐾 ∈ ℕ₀ → (((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2)))))
75	74	impcom 406	. . . . . 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 429	. 2 ⊢ (¬ 1 < 𝐾 → (¬ (𝐾 = 0 ∨ 𝐾 = 2) → ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))))
80		simprl 769	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐺 ∈ FriendGraph )
81		simpl 481	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ∈ Fin)
82	81	ad2antlr 725	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ∈ Fin)
83		simpr 483	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝑉 ≠ ∅)
84	83	ad2antlr 725	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝑉 ≠ ∅)
85		simpl 481	. . . . . 6 ⊢ ((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → 1 < 𝐾)
86	85, 26	anim12ci 612	. . . . 5 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾))
87	6	frgrreggt1 30275	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))
88	87	imp 405	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ (𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾)) → 𝐾 = 2)
89	80, 82, 84, 86, 88	syl31anc 1370	. . . 4 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → 𝐾 = 2)
90	89	olcd 872	. . 3 ⊢ (((1 < 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) → (𝐾 = 0 ∨ 𝐾 = 2))
91	90	exp31 418	. 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 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 ∅c0 4322 class class class wbr 5149 ‘cfv 6549 Fincfn 8964 ℝcr 11139 0cc0 11140 1c1 11141 < clt 11280 ≤ cle 11281 ℕcn 12245 2c2 12300 ℕ₀cn0 12505 ℕ₀^*cxnn0 12577 ♯chash 14325 Vtxcvtx 28881 USGraphcusgr 29034 VtxDegcvtxdg 29351 RegUSGraph crusgr 29442 FriendGraph cfrgr 30140

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-ac2 10488 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-ac 10141 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-rp 13010 df-xadd 13128 df-ico 13365 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-hash 14326 df-word 14501 df-lsw 14549 df-concat 14557 df-s1 14582 df-substr 14627 df-pfx 14657 df-reps 14755 df-csh 14775 df-s2 14835 df-s3 14836 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 df-dvds 16235 df-gcd 16473 df-prm 16646 df-phi 16738 df-vtx 28883 df-iedg 28884 df-edg 28933 df-uhgr 28943 df-ushgr 28944 df-upgr 28967 df-umgr 28968 df-uspgr 29035 df-usgr 29036 df-fusgr 29202 df-nbgr 29218 df-vtxdg 29352 df-rgr 29443 df-rusgr 29444 df-wlks 29485 df-wlkson 29486 df-trls 29578 df-trlson 29579 df-pths 29602 df-spths 29603 df-pthson 29604 df-spthson 29605 df-wwlks 29713 df-wwlksn 29714 df-wwlksnon 29715 df-wspthsn 29716 df-wspthsnon 29717 df-clwwlk 29864 df-clwwlkn 29907 df-clwwlknon 29970 df-frgr 30141

This theorem is referenced by: frgrregord013 30277

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