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Theorem nbhashuvtx1 26442

Description: If the number of the neighbors of a vertex in a finite graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.)

**Assertion**
Ref	Expression
nbhashuvtx1	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))

**Proof of Theorem nbhashuvtx1**
Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
2	1	2a1d 26	. 2 ⊢ (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
3		df-nel 2783	. . 3 ⊢ (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ ¬ 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))
4		nbgrassvwo2 25967	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
5	4	ex 449	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})))
6	5	3ad2ant1 1075	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})))
7		difexg 4735	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑀, 𝑁}) ∈ V)
8	7	3ad2ant2 1076	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑀, 𝑁}) ∈ V)
9		hashss 13058	. . . . . . 7 ⊢ (((𝑉 ∖ {𝑀, 𝑁}) ∈ V ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})))
10	9	ex 449	. . . . . 6 ⊢ ((𝑉 ∖ {𝑀, 𝑁}) ∈ V → ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁}))))
11	8, 10	syl 17	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁}))))
12		simpl2 1058	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → 𝑉 ∈ Fin)
13		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ 𝑉)
14		simp3 1056	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
15		prssi 4293	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → {𝑀, 𝑁} ⊆ 𝑉)
16	13, 14, 15	syl2anr 494	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → {𝑀, 𝑁} ⊆ 𝑉)
17		hashssdif 13061	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ {𝑀, 𝑁} ⊆ 𝑉) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − (#‘{𝑀, 𝑁})))
18	12, 16, 17	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − (#‘{𝑀, 𝑁})))
19		simprr 792	. . . . . . . . . . . . 13 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → 𝑀 ≠ 𝑁)
20		hashprgOLD 13044	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑀 ≠ 𝑁 ↔ (#‘{𝑀, 𝑁}) = 2))
21	13, 14, 20	syl2anr 494	. . . . . . . . . . . . 13 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → (𝑀 ≠ 𝑁 ↔ (#‘{𝑀, 𝑁}) = 2))
22	19, 21	mpbid 221	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → (#‘{𝑀, 𝑁}) = 2)
23	22	oveq2d 6565	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘𝑉) − (#‘{𝑀, 𝑁})) = ((#‘𝑉) − 2))
24	18, 23	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → (#‘(𝑉 ∖ {𝑀, 𝑁})) = ((#‘𝑉) − 2))
25	24	breq2d 4595	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
26		nbhashnn0 26441	. . . . . . . . . . . . . 14 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℕ₀)
27	26	nn0zd 11356	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℤ)
28		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ₀)
29		nn0z 11277	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℕ₀ → (#‘𝑉) ∈ ℤ)
30		peano2zm 11297	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℤ → ((#‘𝑉) − 1) ∈ ℤ)
31	28, 29, 30	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) − 1) ∈ ℤ)
32	31	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘𝑉) − 1) ∈ ℤ)
33		zltlem1 11307	. . . . . . . . . . . . 13 ⊢ (((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℤ ∧ ((#‘𝑉) − 1) ∈ ℤ) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1)))
34	27, 32, 33	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1)))
35	28	nn0cnd 11230	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℂ)
36	35	3ad2ant2 1076	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘𝑉) ∈ ℂ)
37		1cnd 9935	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → 1 ∈ ℂ)
38	36, 37, 37	subsub4d 10302	. . . . . . . . . . . . . 14 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (((#‘𝑉) − 1) − 1) = ((#‘𝑉) − (1 + 1)))
39		1p1e2 11011	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) = 2
40	39	oveq2i 6560	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑉) − (1 + 1)) = ((#‘𝑉) − 2)
41	38, 40	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (((#‘𝑉) − 1) − 1) = ((#‘𝑉) − 2))
42	41	breq2d 4595	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (((#‘𝑉) − 1) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
43	34, 42	bitrd 267	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
44	43	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2)))
45	26	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
46	45	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
47	46	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℝ)
48		simpr 476	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1))
49	47, 48	ltned 10052	. . . . . . . . . . . 12 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1)) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1))
50	49	ex 449	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1)))
51		eqneqall 2793	. . . . . . . . . . . 12 ⊢ ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
52	51	com12 32	. . . . . . . . . . 11 ⊢ ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≠ ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
53	50, 52	syl6 34	. . . . . . . . . 10 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) < ((#‘𝑉) − 1) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
54	44, 53	sylbird 249	. . . . . . . . 9 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ ((#‘𝑉) − 2) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
55	25, 54	sylbid 229	. . . . . . . 8 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁)) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
56	55	ex 449	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
57	56	com23 84	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
58	57	com34 89	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ≤ (#‘(𝑉 ∖ {𝑀, 𝑁})) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
59	6, 11, 58	3syld 58	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
60	59	com12 32	. . 3 ⊢ (𝑀 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
61	3, 60	sylbir 224	. 2 ⊢ (¬ 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))))
62	2, 61	pm2.61i 175	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 #chash 12979 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-2o 7448 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-nbgra 25949 df-vdgr 26421

This theorem is referenced by: nbhashuvtx 26443

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