Theorem vdgn1frgrav2 26553

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, 21-Dec-2017.)

Assertion

Ref	Expression
vdgn1frgrav2	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrav2

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

Step	Hyp	Ref	Expression
1		frisusgra 26519	. . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸)
2	1	anim1i 590	. . . . 5 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉))
3	2	adantr 480	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉))
4		vdusgraval 26434	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
5	3, 4	syl 17	. . 3 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
6		3cyclfrgrarn2 26541	. . . . . 6 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
7	6	adantlr 747	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
8		preq1 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
9	8	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
10		preq2 4213	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
11	10	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
12	9, 11	3anbi13d 1393	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
13	12	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑁 → ((𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
14	13	2rexbidv 3039	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑁 → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
15	14	rspcva 3280	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
16	1	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑉 USGrph 𝐸)
17		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑁 ∈ 𝑉)
18		simplll 794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑏 ≠ 𝑐)
19		3simpb 1052	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
20	19	ad3antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
21		usgra2edg1 25912	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ 𝑏 ≠ 𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
22	16, 17, 18, 20, 21	syl31anc 1321	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
23	22	a1d 25	. . . . . . . . . . . . . . 15 ⊢ ((((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) ∧ 𝑉 FriendGrph 𝐸) → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))
24	23	ex 449	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁 ∈ 𝑉) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))
25	24	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → ((𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
27	26	rexlimivv 3018	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
28	15, 27	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
29	28	ex 449	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁 ∈ 𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
30	29	pm2.43a 52	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
31	30	com24 93	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (1 < (#‘𝑉) → (𝑉 FriendGrph 𝐸 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
32	31	com3r 85	. . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → (𝑁 ∈ 𝑉 → (1 < (#‘𝑉) → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
33	32	imp31 447	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))
34	7, 33	mpd 15	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
35		usgrav 25867	. . . . . . . . . . 11 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
36	35	simprd 478	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → 𝐸 ∈ V)
37		dmexg 6989	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → dom 𝐸 ∈ V)
38	1, 36, 37	3syl 18	. . . . . . . . 9 ⊢ (𝑉 FriendGrph 𝐸 → dom 𝐸 ∈ V)
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → dom 𝐸 ∈ V)
40	39	adantr 480	. . . . . . 7 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ V)
41		rabexg 4739	. . . . . . 7 ⊢ (dom 𝐸 ∈ V → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ V)
42		hash1snb 13068	. . . . . . 7 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = {𝑖}))
43	40, 41, 42	3syl 18	. . . . . 6 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = {𝑖}))
44		reusn 4206	. . . . . 6 ⊢ (∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = {𝑖})
45	43, 44	syl6bbr 277	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))
46	45	necon3abid 2818	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))
47	34, 46	mpbird 246	. . 3 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≠ 1)
48	5, 47	eqnetrd 2849	. 2 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1)
49	48	ex 449	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1c1 9816   < clt 9953  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-vdgr 26421  df-frgra 26516

This theorem is referenced by:  vdgfrgragt2  26554  vdgn1frgrav3  26555