Theorem hausnei2 23671

Description: The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)

Assertion

Ref	Expression
hausnei2	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))

Distinct variable groups:   𝑥,𝑦   𝑣,𝑢,𝑥,𝑦,𝐽   𝑢,𝑋,𝑣,𝑥,𝑦

Proof of Theorem hausnei2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishaus2 23669	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
2		topontop 23231	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝐽 ∈ Top)
4		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑚 ∈ 𝐽)
5		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑥 ∈ 𝑚)
6		opnneip 23437	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
7	3, 4, 5, 6	syl2an3an 1449	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
8		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑛 ∈ 𝐽)
9		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑦 ∈ 𝑛)
10		opnneip 23437	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
11	3, 8, 9, 10	syl2an3an 1449	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
12		simpr3 1215	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
13		ineq1 4159	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∩ 𝑣) = (𝑚 ∩ 𝑣))
14	13	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑣) = ∅))
15		ineq2 4160	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑛 → (𝑚 ∩ 𝑣) = (𝑚 ∩ 𝑛))
16	15	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑛 → ((𝑚 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑛) = ∅))
17	14, 16	rspc2ev 3589	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑦}) ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)
18	7, 11, 12, 17	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)
19	18	ex 418	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
20	19	3expib 1140	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
21	20	rexlimdvv 3219	. . . . . 6 ⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
22		neii2 23426	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
23	22	ex 418	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑢 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)))
24		neii2 23426	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
25	24	ex 418	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘{𝑦}) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
26		vex 3455	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26	snss 4745	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑚 ↔ {𝑥} ⊆ 𝑚)
28	27	anbi1i 636	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ↔ ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
29		vex 3455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
30	29	snss 4745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑛 ↔ {𝑦} ⊆ 𝑛)
31	30	anbi1i 636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
32		simp1l 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 ∈ 𝑚)
33		simp2l 1218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑦 ∈ 𝑛)
34		ss2in 4190	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → (𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣))
35		ssn0 4355	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) ∧ (𝑚 ∩ 𝑛) ≠ ∅) → (𝑢 ∩ 𝑣) ≠ ∅)
36	35	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑚 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑣) ≠ ∅))
37	36	necon4d 2980	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
38	34, 37	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
39	38	ad2ant2l 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
40	39	3impia 1135	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑚 ∩ 𝑛) = ∅)
41	32, 33, 40	3jca 1146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
42	41	3exp 1137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
43	31, 42	biimtrrid 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
44	43	com3r 88	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
45	44	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
46	45	3adant1 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
47	46	reximdv 3178	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
48	47	3exp 1137	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
49	48	com34 92	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
50	49	3imp 1128	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
51	28, 50	biimtrrid 246	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
52	51	reximdv 3178	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
53	52	3exp 1137	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
54	53	com24 96	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
55	54	impd 416	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
56	23, 25, 55	syl2and 620	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑢 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
57	56	rexlimdvv 3219	. . . . . 6 ⊢ (𝐽 ∈ Top → (∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
58	21, 57	impbid 215	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
59	58	imbi2d 343	. . . 4 ⊢ (𝐽 ∈ Top → ((𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
60	59	2ralbidv 3227	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
61	2, 60	syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
62	1, 61	bitrd 282	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ‘cfv 6538  Topctop 23211  TopOnctopon 23228  neicnei 23415  Hauscha 23626

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23212  df-topon 23229  df-nei 23416  df-haus 23633

This theorem is used by:  hausflim  24300