Theorem hausnei2 21958

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 21956	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
2		topontop 21518	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝐽 ∈ Top)
4		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑚 ∈ 𝐽)
5		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑥 ∈ 𝑚)
6		opnneip 21724	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
7	3, 4, 5, 6	syl2an3an 1419	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
8		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑛 ∈ 𝐽)
9		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑦 ∈ 𝑛)
10		opnneip 21724	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
11	3, 8, 9, 10	syl2an3an 1419	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
12		simpr3 1193	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
13		ineq1 4131	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∩ 𝑣) = (𝑚 ∩ 𝑣))
14	13	eqeq1d 2800	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑣) = ∅))
15		ineq2 4133	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑛 → (𝑚 ∩ 𝑣) = (𝑚 ∩ 𝑛))
16	15	eqeq1d 2800	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑛 → ((𝑚 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑛) = ∅))
17	14, 16	rspc2ev 3583	. . . . . . . . . 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 1368	. . . . . . . . 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 416	. . . . . . . 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 1119	. . . . . . 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 3252	. . . . . 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 21713	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
23	22	ex 416	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑢 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)))
24		neii2 21713	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
25	24	ex 416	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘{𝑦}) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
26		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26	snss 4679	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑚 ↔ {𝑥} ⊆ 𝑚)
28	27	anbi1i 626	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ↔ ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
29		vex 3444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
30	29	snss 4679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑛 ↔ {𝑦} ⊆ 𝑛)
31	30	anbi1i 626	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
32		simp1l 1194	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 ∈ 𝑚)
33		simp2l 1196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑦 ∈ 𝑛)
34		ss2in 4163	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → (𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣))
35		ssn0 4308	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) ∧ (𝑚 ∩ 𝑛) ≠ ∅) → (𝑢 ∩ 𝑣) ≠ ∅)
36	35	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑚 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑣) ≠ ∅))
37	36	necon4d 3011	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
38	34, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
39	38	ad2ant2l 745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
40	39	3impia 1114	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑚 ∩ 𝑛) = ∅)
41	32, 33, 40	3jca 1125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
42	41	3exp 1116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
43	31, 42	syl5bir 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
44	43	com3r 87	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
45	44	imp 410	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
46	45	3adant1 1127	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
47	46	reximdv 3232	. . . . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . 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 91	. . . . . . . . . . . . . 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 1108	. . . . . . . . . . . . 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	syl5bir 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 3232	. . . . . . . . . . 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 1116	. . . . . . . . . 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 95	. . . . . . . . 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 414	. . . . . . . 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 610	. . . . . . 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 3252	. . . . . 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 344	. . . 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 3164	. . 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 17	. 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) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  ‘cfv 6324  Topctop 21498  TopOnctopon 21515  neicnei 21702  Hauscha 21913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-topon 21516  df-nei 21703  df-haus 21920

This theorem is referenced by:  hausflim  22586