Theorem dfsclnbgr6 47838

Description: Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (Contributed by AV, 17-May-2025.)

Hypotheses

Ref	Expression
dfvopnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u	⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
dfsclnbgr6.s	⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}

Assertion

Ref	Expression
dfsclnbgr6	⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸   𝑛,𝐺

Allowed substitution hints:   𝑆(𝑒,𝑛)   𝑈(𝑒,𝑛)

Proof of Theorem dfsclnbgr6

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → 𝑛 ∈ 𝑒)
2	1	anim1i 615	. . . . . . . . . 10 ⊢ (((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∧ 𝑛 = 𝑁) → (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))
3	2	olcd 874	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∧ 𝑛 = 𝑁) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
4	3	expcom 413	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
5		3anass 1094	. . . . . . . . . . . 12 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
6	5	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
7	6	orcd 873	. . . . . . . . . 10 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})))
8	7	orcd 873	. . . . . . . . 9 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
9	8	ex 412	. . . . . . . 8 ⊢ (𝑛 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
10	4, 9	pm2.61ine 3016	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
11		3simpc 1150	. . . . . . . . . 10 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
13		vsnid 4644	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ {𝑛}
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑛} → 𝑛 ∈ {𝑛})
15		eleq2 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑛} → (𝑛 ∈ 𝑒 ↔ 𝑛 ∈ {𝑛}))
16	14, 15	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑛} → 𝑛 ∈ 𝑒)
17	16	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → 𝑛 ∈ 𝑒)
18		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ 𝑒 ↔ 𝑁 ∈ 𝑒))
19	18	bicomd 223	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑁 ∈ 𝑒 ↔ 𝑛 ∈ 𝑒))
20	19	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → (𝑁 ∈ 𝑒 ↔ 𝑛 ∈ 𝑒))
21	17, 20	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → 𝑁 ∈ 𝑒)
22	21	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → 𝑁 ∈ 𝑒)
23	17	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → 𝑛 ∈ 𝑒)
24	22, 23	jca 511	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
25	24	ex 412	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
26	12, 25	jaod 859	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
27	18	biimpac 478	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → 𝑁 ∈ 𝑒)
28		simpl 482	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → 𝑛 ∈ 𝑒)
29	27, 28	jca 511	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
30	29	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
31	26, 30	jaod 859	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → ((((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
32	10, 31	impbid2 226	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
33	32	rexbidv 3165	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
34		r19.43 3109	. . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
35	34	a1i 11	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
36		r19.41v 3175	. . . . . . . 8 ⊢ (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))
37	36	biancomi 462	. . . . . . 7 ⊢ (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))
38	37	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)))
39	38	orbi2d 915	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → ((∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))))
40	33, 35, 39	3bitrd 305	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))))
41	40	rabbidv 3428	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))})
42		unrab 4295	. . . 4 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)}) = {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))}
43		rabsneq 4625	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒} = {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)})
44	43	eqcomd 2742	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)} = {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒})
45	44	uneq2d 4148	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
46	42, 45	eqtr3id 2785	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
47	41, 46	eqtrd 2771	. 2 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
48		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
49		dfsclnbgr6.s	. . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
50		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
51	48, 49, 50	dfsclnbgr2 47826	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
52		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
53	48, 50, 52	dfvopnbgr2 47833	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
54	53	uneq1d 4147	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
55	47, 51, 54	3eqtr4d 2781	1 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  {crab 3420   ∪ cun 3929   ⊆ wss 3931  {csn 4606  {cpr 4608  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 28980  Edgcedg 29031   NeighbVtx cnbgr 29316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-nbgr 29317

This theorem is referenced by:  dfnbgrss2  47839