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Theorem hausflim 23924

Description: A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

**Hypothesis**
Ref	Expression
flimcf.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
hausflim	⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))

Distinct variable groups: 𝑥,𝑓,𝐽 𝑓,𝑋,𝑥

Proof of Theorem hausflim Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 23274	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2		hausflimi 23923	. . . 4 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
3	2	ralrimivw 3137	. . 3 ⊢ (𝐽 ∈ Haus → ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
4	1, 3	jca 511	. 2 ⊢ (𝐽 ∈ Haus → (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
5		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝐽 ∈ Top)
6		flimcf.1	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = ∪ 𝐽
7	6	toptopon 22860	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝐽 ∈ (TopOn‘𝑋))
9		simprll 778	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑧 ∈ 𝑋)
10	9	snssd 4790	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ⊆ 𝑋)
11	9	snn0d 4756	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ≠ ∅)
12		neifil 23823	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑧} ⊆ 𝑋 ∧ {𝑧} ≠ ∅) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
13	8, 10, 11, 12	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
14		filfbas 23791	. . . . . . . . . . . 12 ⊢ (((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋))
15	13, 14	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋))
16		simprlr 779	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑤 ∈ 𝑋)
17	16	snssd 4790	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ⊆ 𝑋)
18	16	snn0d 4756	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ≠ ∅)
19		neifil 23823	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑤} ⊆ 𝑋 ∧ {𝑤} ≠ ∅) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
20	8, 17, 18, 19	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
21		filfbas 23791	. . . . . . . . . . . 12 ⊢ (((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋))
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋))
23		fbunfip 23812	. . . . . . . . . . 11 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋) ∧ ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅))
24	15, 22, 23	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅))
25	6	neisspw 23050	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑧}) ⊆ 𝒫 𝑋)
26	6	neisspw 23050	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑤}) ⊆ 𝒫 𝑋)
27	25, 26	unssd 4172	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋)
28	27	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋)
29	28	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋))
30		ssun1 4158	. . . . . . . . . . . . . 14 ⊢ ((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))
31		filn0 23805	. . . . . . . . . . . . . . 15 ⊢ (((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ≠ ∅)
32	13, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ≠ ∅)
33		ssn0 4384	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∧ ((nei‘𝐽)‘{𝑧}) ≠ ∅) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)
34	30, 32, 33	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)
35	34	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅))
36		idd 24	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
37	29, 35, 36	3jcad 1129	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
38	6	topopn 22849	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑋 ∈ 𝐽)
40		fsubbas 23810	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
42		fgcl 23821	. . . . . . . . . . . . . . 15 ⊢ ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋))
43	42	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋))
44		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ≠ 𝑤)
45	9	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ 𝑋)
46	16	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ 𝑋)
47		fvex 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑧}) ∈ V
48		fvex 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑤}) ∈ V
49	47, 48	unex 7743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V
50		ssfii 9436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))
51	49, 50	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))
52		ssfg 23815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
54	51, 53	sstrid 3975	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
55	30, 54	sstrid 3975	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
56	8	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
57		elflim 23914	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
58	56, 43, 57	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
59	45, 55, 58	mpbir2and 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
60	54	unssbd 4174	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
61		elflim 23914	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
62	56, 43, 61	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
63	46, 60, 62	mpbir2and 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
64		eleq1w 2818	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
65		eleq1w 2818	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
66	64, 65	moi 3706	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → 𝑧 = 𝑤)
67	66	3com23 1126	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → 𝑧 = 𝑤)
68	67	3expia 1121	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤))
69	45, 46, 59, 63, 68	syl22anc 838	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤))
70	69	necon3ad 2946	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ≠ 𝑤 → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
71	44, 70	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
72		oveq2 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝐽 fLim 𝑓) = (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
73	72	eleq2d 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
74	73	mobidv 2549	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
75	74	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
76	75	rspcev 3606	. . . . . . . . . . . . . 14 ⊢ (((𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋) ∧ ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
77	43, 71, 76	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
78	77	ex 412	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
79	41, 78	sylbird 260	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
80	37, 79	syld 47	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
81	24, 80	sylbird 260	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
82		df-ne 2934	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ (𝑢 ∩ 𝑣) = ∅)
83	82	ralbii 3083	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅)
84		ralnex 3063	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
85	83, 84	bitri 275	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
86	85	ralbii 3083	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
87		ralnex 3063	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
88	86, 87	bitri 275	. . . . . . . . 9 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
89		rexnal 3090	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (Fil‘𝑋) ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∀𝑓 ∈ (Fil‘𝑋)∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
90	81, 88, 89	3imtr3g 295	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ → ¬ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
91	90	con4d 115	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
92	91	imp 406	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
93	92	an32s 652	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
94	93	expr 456	. . . 4 ⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
95	94	ralrimivva 3188	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
96		simpl 482	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ Top)
97	96, 7	sylib 218	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
98		hausnei2 23296	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)))
99	97, 98	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)))
100	95, 99	mpbird 257	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ Haus)
101	4, 100	impbii 209	1 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃*wmo 2538 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 Vcvv 3464 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 𝒫 cpw 4580 {csn 4606 ∪ cuni 4888 ‘cfv 6536 (class class class)co 7410 ficfi 9427 fBascfbas 21308 filGencfg 21309 Topctop 22836 TopOnctopon 22853 neicnei 23040 Hauscha 23251 Filcfil 23788 fLim cflim 23877

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-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 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-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 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-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1o 8485 df-2o 8486 df-en 8965 df-fin 8968 df-fi 9428 df-fbas 21317 df-fg 21318 df-top 22837 df-topon 22854 df-nei 23041 df-haus 23258 df-fil 23789 df-flim 23882

This theorem is referenced by: (None)

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