Theorem vdgn1frgrv2 30232

Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
vdn1frgrv2.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
vdgn1frgrv2	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrv2

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrusgr 30197	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2	1	anim1i 615	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
3	2	adantr 480	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
4		vdn1frgrv2.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2730	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2730	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7		eqid 2730	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
8	4, 5, 6, 7	vtxdusgrval 29422	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
9	3, 8	syl 17	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
10		eqid 2730	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
11	4, 10	3cyclfrgrrn2 30223	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
12	11	adantlr 715	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
13		preq1 4700	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
14	13	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15		preq2 4701	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
16	15	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
17	14, 16	3anbi13d 1440	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)) ↔ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
18	17	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑁 → ((𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
19	18	2rexbidv 3203	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑁 → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
20	19	rspcva 3589	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
21	1	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ USGraph)
22		simplll 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑏 ≠ 𝑐)
23		3simpb 1149	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
24	23	ad3antlr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
25	5, 10	usgr2edg1 29146	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USGraph ∧ 𝑏 ≠ 𝑐) ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
26	21, 22, 24, 25	syl21anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27	26	a1d 25	. . . . . . . . . . . . . . 15 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
28	27	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))
29	28	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁 ∈ 𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁 ∈ 𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
31	30	rexlimivv 3180	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁 ∈ 𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
32	20, 31	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → (𝑁 ∈ 𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
33	32	ex 412	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝑁 ∈ 𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
34	33	pm2.43a 54	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
35	34	com24 95	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
36	35	com3r 87	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → (𝑁 ∈ 𝑉 → (1 < (♯‘𝑉) → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
37	36	imp31 417	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
38	12, 37	mpd 15	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
39		fvex 6874	. . . . . . . . 9 ⊢ (iEdg‘𝐺) ∈ V
40	39	dmex 7888	. . . . . . . 8 ⊢ dom (iEdg‘𝐺) ∈ V
41	40	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42		rabexg 5295	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43		hash1snb 14391	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
44	41, 42, 43	3syl 18	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
45		reusn 4694	. . . . . 6 ⊢ (∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖})
46	44, 45	bitr4di 289	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
47	46	necon3abid 2962	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
48	38, 47	mpbird 257	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1)
49	9, 48	eqnetrd 2993	. 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1)
50	49	ex 412	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354  {crab 3408  Vcvv 3450  {csn 4592  {cpr 4594   class class class wbr 5110  dom cdm 5641  ‘cfv 6514  1c1 11076   < clt 11215  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  Edgcedg 28981  USGraphcusgr 29083  VtxDegcvtxdg 29400   FriendGraph cfrgr 30194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-xadd 13080  df-fz 13476  df-hash 14303  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-umgr 29017  df-usgr 29085  df-vtxdg 29401  df-frgr 30195

This theorem is referenced by:  vdgn1frgrv3  30233  vdgfrgrgt2  30234