Theorem vdgn1frgrv2 28796

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 28761	. . . . . 6 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2	1	anim1i 615	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
3	2	adantr 481	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
4		vdn1frgrv2.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2737	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2737	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7		eqid 2737	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
8	4, 5, 6, 7	vtxdusgrval 27990	. . . 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 2737	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
11	4, 10	3cyclfrgrrn2 28787	. . . . . 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 712	. . . . 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 4679	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
14	13	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15		preq2 4680	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
16	15	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
17	14, 16	3anbi13d 1437	. . . . . . . . . . . . . 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 629	. . . . . . . . . . . . 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 3210	. . . . . . . . . . . 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 3568	. . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ USGraph)
22		simplll 772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑏 ≠ 𝑐)
23		3simpb 1148	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
24	23	ad3antlr 728	. . . . . . . . . . . . . . . . 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 27715	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 3193	. . . . . . . . . . 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 413	. . . . . . . . 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 418	. . . . 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 6825	. . . . . . . . 9 ⊢ (iEdg‘𝐺) ∈ V
40	39	dmex 7805	. . . . . . . 8 ⊢ dom (iEdg‘𝐺) ∈ V
41	40	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42		rabexg 5270	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43		hash1snb 14213	. . . . . . 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 4673	. . . . . 6 ⊢ (∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖})
46	44, 45	bitr4di 288	. . . . 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 2978	. . . 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 256	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1)
49	9, 48	eqnetrd 3009	. 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1)
50	49	ex 413	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  ∃!wreu 3348  {crab 3404  Vcvv 3441  {csn 4571  {cpr 4573   class class class wbr 5087  dom cdm 5608  ‘cfv 6466  1c1 10952   < clt 11089  ♯chash 14124  Vtxcvtx 27502  iEdgciedg 27503  Edgcedg 27553  USGraphcusgr 27655  VtxDegcvtxdg 27968   FriendGraph cfrgr 28758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-oadd 8350  df-er 8548  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-dju 9737  df-card 9775  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-2 12116  df-n0 12314  df-xnn0 12386  df-z 12400  df-uz 12663  df-xadd 12929  df-fz 13320  df-hash 14125  df-edg 27554  df-uhgr 27564  df-upgr 27588  df-umgr 27589  df-usgr 27657  df-vtxdg 27969  df-frgr 28759

This theorem is referenced by:  vdgn1frgrv3  28797  vdgfrgrgt2  28798