Theorem vdgn1frgrv2 29587

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 29552	. . . . . 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 2732	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6		eqid 2732	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7		eqid 2732	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
8	4, 5, 6, 7	vtxdusgrval 28782	. . . 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 2732	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
11	4, 10	3cyclfrgrrn2 29578	. . . . . 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 713	. . . . 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 4737	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
14	13	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15		preq2 4738	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
16	15	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
17	14, 16	3anbi13d 1438	. . . . . . . . . . . . . 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 3219	. . . . . . . . . . . 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 3610	. . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . 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 28507	. . . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . . 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 3199	. . . . . . . . . . 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 6904	. . . . . . . . 9 ⊢ (iEdg‘𝐺) ∈ V
40	39	dmex 7904	. . . . . . . 8 ⊢ dom (iEdg‘𝐺) ∈ V
41	40	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑁 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42		rabexg 5331	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43		hash1snb 14381	. . . . . . 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 4731	. . . . . 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 2977	. . . 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 3008	. 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 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∃!wreu 3374  {crab 3432  Vcvv 3474  {csn 4628  {cpr 4630   class class class wbr 5148  dom cdm 5676  ‘cfv 6543  1c1 11113   < clt 11250  ♯chash 14292  Vtxcvtx 28294  iEdgciedg 28295  Edgcedg 28345  USGraphcusgr 28447  VtxDegcvtxdg 28760   FriendGraph cfrgr 29549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-xadd 13095  df-fz 13487  df-hash 14293  df-edg 28346  df-uhgr 28356  df-upgr 28380  df-umgr 28381  df-usgr 28449  df-vtxdg 28761  df-frgr 29550

This theorem is referenced by:  vdgn1frgrv3  29588  vdgfrgrgt2  29589