Theorem hausnei2 23478

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 23476	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
2		topontop 23038	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3		simp1 1152	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝐽 ∈ Top)
4		simp2 1153	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑚 ∈ 𝐽)
5		simp1 1152	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑥 ∈ 𝑚)
6		opnneip 23244	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
7	3, 4, 5, 6	syl2an3an 1447	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
8		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑛 ∈ 𝐽)
9		simp2 1153	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑦 ∈ 𝑛)
10		opnneip 23244	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
11	3, 8, 9, 10	syl2an3an 1447	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
12		simpr3 1213	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
13		ineq1 4174	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∩ 𝑣) = (𝑚 ∩ 𝑣))
14	13	eqeq1d 2771	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑣) = ∅))
15		ineq2 4175	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑛 → (𝑚 ∩ 𝑣) = (𝑚 ∩ 𝑛))
16	15	eqeq1d 2771	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑛 → ((𝑚 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑛) = ∅))
17	14, 16	rspc2ev 3603	. . . . . . . . . 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 1396	. . . . . . . . 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 417	. . . . . . . 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 1138	. . . . . . 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 3227	. . . . . 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 23233	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
23	22	ex 417	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑢 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)))
24		neii2 23233	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
25	24	ex 417	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘{𝑦}) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
26		vex 3467	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26	snss 4755	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑚 ↔ {𝑥} ⊆ 𝑚)
28	27	anbi1i 635	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ↔ ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
29		vex 3467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
30	29	snss 4755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑛 ↔ {𝑦} ⊆ 𝑛)
31	30	anbi1i 635	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
32		simp1l 1214	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 ∈ 𝑚)
33		simp2l 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑦 ∈ 𝑛)
34		ss2in 4205	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → (𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣))
35		ssn0 4368	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) ∧ (𝑚 ∩ 𝑛) ≠ ∅) → (𝑢 ∩ 𝑣) ≠ ∅)
36	35	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑚 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑣) ≠ ∅))
37	36	necon4d 2988	. . . . . . . . . . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
40	39	3impia 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑚 ∩ 𝑛) = ∅)
41	32, 33, 40	3jca 1144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
42	41	3exp 1135	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
46	45	3adant1 1146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
47	46	reximdv 3186	. . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . 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 3186	. . . . . . . . . . 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 1135	. . . . . . . . . 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 415	. . . . . . . 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 619	. . . . . . 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 3227	. . . . . 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 3235	. . 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) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594  ‘cfv 6537  Topctop 23018  TopOnctopon 23035  neicnei 23222  Hauscha 23433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23019  df-topon 23036  df-nei 23223  df-haus 23440

This theorem is referenced by:  hausflim  24106