Theorem hausnei2 20967

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 20965	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥 ≠ 𝑦 → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
2		topontop 20541	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3		simp1 1054	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝐽 ∈ Top)
4	3	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝐽 ∈ Top)
5		simp2 1055	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑚 ∈ 𝐽)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ 𝐽)
7		simp1 1054	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑥 ∈ 𝑚)
8	7	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑥 ∈ 𝑚)
9		opnneip 20733	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
10	4, 6, 8, 9	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑚 ∈ ((nei‘𝐽)‘{𝑥}))
11		simp3 1056	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → 𝑛 ∈ 𝐽)
12	11	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ 𝐽)
13		simp2 1055	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → 𝑦 ∈ 𝑛)
14	13	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑦 ∈ 𝑛)
15		opnneip 20733	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽 ∧ 𝑦 ∈ 𝑛) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
16	4, 12, 14, 15	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → 𝑛 ∈ ((nei‘𝐽)‘{𝑦}))
17		simp3 1056	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → (𝑚 ∩ 𝑛) = ∅)
18	17	adantl 481	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
19		ineq1 3769	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∩ 𝑣) = (𝑚 ∩ 𝑣))
20	19	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑣) = ∅))
21		ineq2 3770	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑛 → (𝑚 ∩ 𝑣) = (𝑚 ∩ 𝑛))
22	21	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑛 → ((𝑚 ∩ 𝑣) = ∅ ↔ (𝑚 ∩ 𝑛) = ∅))
23	20, 22	rspc2ev 3295	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑦}) ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)
24	10, 16, 18, 23	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) ∧ (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)
25	24	ex 449	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
26	25	3expib 1260	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑚 ∈ 𝐽 ∧ 𝑛 ∈ 𝐽) → ((𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅)))
27	26	rexlimdvv 3019	. . . . . 6 ⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
28		neii2 20722	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
29	28	ex 449	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑢 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑚 ∈ 𝐽 ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢)))
30		neii2 20722	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
31	30	ex 449	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘{𝑦}) → ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
32		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
33	32	snss 4259	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑚 ↔ {𝑥} ⊆ 𝑚)
34	33	anbi1i 727	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ↔ ({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢))
35		vex 3176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑦 ∈ V
36	35	snss 4259	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑛 ↔ {𝑦} ⊆ 𝑛)
37	36	anbi1i 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣))
38		simp1l 1078	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑥 ∈ 𝑚)
39		simp2l 1080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → 𝑦 ∈ 𝑛)
40		ss2in 3802	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → (𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣))
41		ssn0 3928	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) ∧ (𝑚 ∩ 𝑛) ≠ ∅) → (𝑢 ∩ 𝑣) ≠ ∅)
42	41	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑚 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑣) ≠ ∅))
43	42	necon4d 2806	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∩ 𝑛) ⊆ (𝑢 ∩ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
44	40, 43	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ⊆ 𝑢 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
45	44	ad2ant2l 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑢 ∩ 𝑣) = ∅ → (𝑚 ∩ 𝑛) = ∅))
46	45	3impia 1253	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑚 ∩ 𝑛) = ∅)
47	38, 39, 46	3jca 1235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) ∧ (𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ∧ (𝑢 ∩ 𝑣) = ∅) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))
48	47	3exp 1256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ((𝑦 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
49	37, 48	syl5bir 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑢 ∩ 𝑣) = ∅ → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
50	49	com3r 85	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
51	50	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
52	51	3adant1 1072	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
53	52	reximdv 2999	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ (𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢)) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
54	53	3exp 1256	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
55	54	com34 89	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → ((𝑢 ∩ 𝑣) = ∅ → (∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))))
56	55	3imp 1249	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → ((𝑥 ∈ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
57	34, 56	syl5bir 232	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝑣) = ∅ ∧ ∃𝑛 ∈ 𝐽 ({𝑦} ⊆ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (({𝑥} ⊆ 𝑚 ∧ 𝑚 ⊆ 𝑢) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
58	57	reximdv 2999	. . . . . . . . . . 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) = ∅)))
59	58	3exp 1256	. . . . . . . . . 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) = ∅)))))
60	59	com24 93	. . . . . . . . 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) = ∅)))))
61	60	impd 446	. . . . . . . 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) = ∅))))
62	29, 31, 61	syl2and 499	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑢 ∈ ((nei‘𝐽)‘{𝑥}) ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑦})) → ((𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅))))
63	62	rexlimdvv 3019	. . . . . 6 ⊢ (𝐽 ∈ Top → (∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅ → ∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅)))
64	27, 63	impbid 201	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑚 ∈ 𝐽 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑚 ∧ 𝑦 ∈ 𝑛 ∧ (𝑚 ∩ 𝑛) = ∅) ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑥})∃𝑣 ∈ ((nei‘𝐽)‘{𝑦})(𝑢 ∩ 𝑣) = ∅))
65	64	imbi2d 329	. . . 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) = ∅)))
66	65	2ralbidv 2972	. . 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) = ∅)))
67	2, 66	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) = ∅)))
68	1, 67	bitrd 267	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ‘cfv 5804  Topctop 20517  TopOnctopon 20518  neicnei 20711  Hauscha 20922

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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-reu 2903  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-topon 20523  df-nei 20712  df-haus 20929

This theorem is referenced by:  hausflim  21595