Theorem nb3graprlem1 25980

Description: Lemma 1 for nb3grapr 25982. (Contributed by Alexander van der Vekens, 15-Oct-2017.)

Assertion

Ref	Expression
nb3graprlem1	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))

Proof of Theorem nb3graprlem1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prid1g 4239	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐵, 𝐶})
2	1	3ad2ant2 1076	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐵 ∈ {𝐵, 𝐶})
3	2	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐵 ∈ {𝐵, 𝐶})
4	3	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ {𝐵, 𝐶})
5		eleq2 2677	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (⟨𝑉, 𝐸⟩ Neighbors 𝐴) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
6	5	eqcoms 2618	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
7	6	adantl 481	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ {𝐵, 𝐶} ↔ 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
8	4, 7	mpbid 221	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴))
9		nbgraeledg 25959	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐵, 𝐴} ∈ ran 𝐸))
10		prcom 4211	. . . . . . . . 9 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
11	10	a1i 11	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → {𝐵, 𝐴} = {𝐴, 𝐵})
12	11	eleq1d 2672	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → ({𝐵, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
13	9, 12	bitrd 267	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
14	13	adantl 481	. . . . 5 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
15	14	ad2antlr 759	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐵 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐵} ∈ ran 𝐸))
16	8, 15	mpbid 221	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐵} ∈ ran 𝐸)
17		prid2g 4240	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐵, 𝐶})
18	17	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ {𝐵, 𝐶})
19	18	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐶 ∈ {𝐵, 𝐶})
20	19	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ {𝐵, 𝐶})
21		eleq2 2677	. . . . . . 7 ⊢ ({𝐵, 𝐶} = (⟨𝑉, 𝐸⟩ Neighbors 𝐴) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
22	21	eqcoms 2618	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
23	22	adantl 481	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ {𝐵, 𝐶} ↔ 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴)))
24	20, 23	mpbid 221	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → 𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴))
25		nbgraeledg 25959	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐶, 𝐴} ∈ ran 𝐸))
26		prcom 4211	. . . . . . . . 9 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
27	26	a1i 11	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → {𝐶, 𝐴} = {𝐴, 𝐶})
28	27	eleq1d 2672	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸))
29	25, 28	bitrd 267	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
30	29	adantl 481	. . . . 5 ⊢ ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
31	30	ad2antlr 759	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → (𝐶 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) ↔ {𝐴, 𝐶} ∈ ran 𝐸))
32	24, 31	mpbid 221	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → {𝐴, 𝐶} ∈ ran 𝐸)
33	16, 32	jca 553	. 2 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
34		nbusgra 25957	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
35	34	ad2antll 761	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
36	35	adantr 480	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸})
37		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
38	37	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
39	38	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
40		vex 3176	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
41	40	eltp 4177	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
42		usgraedgrn 25910	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → 𝐴 ≠ 𝑣)
43		df-ne 2782	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≠ 𝑣 ↔ ¬ 𝐴 = 𝑣)
44		pm2.24 120	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 = 𝑣 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
45	44	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝐴 → (¬ 𝐴 = 𝑣 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
46	45	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐴 = 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
47	43, 46	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≠ 𝑣 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
48	42, 47	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
49	48	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
50	49	ad2antll 761	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐴 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
52	51	com3r 85	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
53		orc 399	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐵 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
54	53	a1d 25	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
55	54	a1d 25	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
56		olc 398	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐶 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))
57	56	a1d 25	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
58	57	a1d 25	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
59	52, 55, 58	3jaoi 1383	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐴 ∨ 𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
60	41, 59	sylbi 206	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
61	60	com12 32	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
62	39, 61	sylbid 229	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ 𝑉 → ({𝐴, 𝑣} ∈ ran 𝐸 → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶))))
63	62	impd 446	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸) → (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
64		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐵 = 𝐵
65	64	3mix2i 1227	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)
66		eltpg 4174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ 𝑌 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)))
67	65, 66	mpbiri 247	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ 𝑌 → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
68	67	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑌 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
69	68	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
70	69	imp 444	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
71	70	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
72		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐵 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
73	72	bicomd 212	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐵 → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
74	73	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
75	71, 74	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
76	37	bicomd 212	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 = {𝐴, 𝐵, 𝐶} → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
77	76	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
78	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
79	75, 78	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐵) → 𝑣 ∈ 𝑉)
80	79	ex 449	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
81	80	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐵 → 𝑣 ∈ 𝑉))
82	81	impcom 445	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → 𝑣 ∈ 𝑉)
83		preq2 4213	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = 𝑣 → {𝐴, 𝐵} = {𝐴, 𝑣})
84	83	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝑣 → ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
85	84	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐵 → ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
86	85	biimpcd 238	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ ran 𝐸))
87	86	ad2antrl 760	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐵 → {𝐴, 𝑣} ∈ ran 𝐸))
88	87	impcom 445	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐵 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → {𝐴, 𝑣} ∈ ran 𝐸)
89	82, 88	jca 553	. . . . . . . . 9 ⊢ ((𝑣 = 𝐵 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸))
90	89	ex 449	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
91		tpid3g 4248	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ 𝑍 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
92	91	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ 𝑍 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
93	92	3ad2ant3 1077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
94	93	imp 444	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
95	94	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
96		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝐶 → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
97	96	bicomd 212	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝐶 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
98	97	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ {𝐴, 𝐵, 𝐶}))
99	95, 98	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝑣 ∈ {𝐴, 𝐵, 𝐶})
100	77	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → (𝑣 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑣 ∈ 𝑉))
101	99, 100	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ 𝑣 = 𝐶) → 𝑣 ∈ 𝑉)
102	101	ex 449	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
103	102	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐶 → 𝑣 ∈ 𝑉))
104	103	impcom 445	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → 𝑣 ∈ 𝑉)
105		preq2 4213	. . . . . . . . . . . . . . 15 ⊢ (𝐶 = 𝑣 → {𝐴, 𝐶} = {𝐴, 𝑣})
106	105	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝐶 = 𝑣 → ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
107	106	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝐶 → ({𝐴, 𝐶} ∈ ran 𝐸 ↔ {𝐴, 𝑣} ∈ ran 𝐸))
108	107	biimpcd 238	. . . . . . . . . . . 12 ⊢ ({𝐴, 𝐶} ∈ ran 𝐸 → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ ran 𝐸))
109	108	ad2antll 761	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 = 𝐶 → {𝐴, 𝑣} ∈ ran 𝐸))
110	109	impcom 445	. . . . . . . . . 10 ⊢ ((𝑣 = 𝐶 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → {𝐴, 𝑣} ∈ ran 𝐸)
111	104, 110	jca 553	. . . . . . . . 9 ⊢ ((𝑣 = 𝐶 ∧ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸))
112	111	ex 449	. . . . . . . 8 ⊢ (𝑣 = 𝐶 → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
113	90, 112	jaoi 393	. . . . . . 7 ⊢ ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
114	113	com12 32	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣 = 𝐵 ∨ 𝑣 = 𝐶) → (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)))
115	63, 114	impbid 201	. . . . 5 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → ((𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸) ↔ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)))
116	115	abbidv 2728	. . . 4 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)})
117		df-rab 2905	. . . 4 ⊢ {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸} = {𝑣 ∣ (𝑣 ∈ 𝑉 ∧ {𝐴, 𝑣} ∈ ran 𝐸)}
118		dfpr2 4143	. . . 4 ⊢ {𝐵, 𝐶} = {𝑣 ∣ (𝑣 = 𝐵 ∨ 𝑣 = 𝐶)}
119	116, 117, 118	3eqtr4g 2669	. . 3 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → {𝑣 ∈ 𝑉 ∣ {𝐴, 𝑣} ∈ ran 𝐸} = {𝐵, 𝐶})
120	36, 119	eqtrd 2644	. 2 ⊢ ((((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶})
121	33, 120	impbida 873	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  {crab 2900  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946

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-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949

This theorem is referenced by:  nb3grapr  25982  nb3grapr2  25983  nb3gra2nb  25984