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Theorem hausflim 23940

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 23290	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2		hausflimi 23939	. . . 4 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
3	2	ralrimivw 3134	. . 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 22876	. . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑧 ∈ 𝑋)
10	9	snssd 4767	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ⊆ 𝑋)
11	9	snn0d 4734	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ≠ ∅)
12		neifil 23839	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑧} ⊆ 𝑋 ∧ {𝑧} ≠ ∅) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
13	8, 10, 11, 12	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
14		filfbas 23807	. . . . . . . . . . . 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 780	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑤 ∈ 𝑋)
17	16	snssd 4767	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ⊆ 𝑋)
18	16	snn0d 4734	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ≠ ∅)
19		neifil 23839	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑤} ⊆ 𝑋 ∧ {𝑤} ≠ ∅) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
20	8, 17, 18, 19	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
21		filfbas 23807	. . . . . . . . . . . 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 23828	. . . . . . . . . . 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 585	. . . . . . . . . 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 23066	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑧}) ⊆ 𝒫 𝑋)
26	6	neisspw 23066	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑤}) ⊆ 𝒫 𝑋)
27	25, 26	unssd 4146	. . . . . . . . . . . . . 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 4132	. . . . . . . . . . . . . 14 ⊢ ((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))
31		filn0 23821	. . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∧ ((nei‘𝐽)‘{𝑧}) ≠ ∅) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)
34	30, 32, 33	sylancr 588	. . . . . . . . . . . . 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 1130	. . . . . . . . . . 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 22865	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑋 ∈ 𝐽)
40		fsubbas 23826	. . . . . . . . . . . . 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 23837	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑧}) ∈ V
48		fvex 6855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑤}) ∈ V
49	47, 48	unex 7699	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V
50		ssfii 9334	. . . . . . . . . . . . . . . . . . . . 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 23831	. . . . . . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . 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 23930	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . . . . 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 23930	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . 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 3678	. . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . 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 1122	. . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . 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 2550	. . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 3084	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅)
84		ralnex 3064	. . . . . . . . . . . 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 3084	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
87		ralnex 3064	. . . . . . . . . 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 653	. . . . 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 3181	. . 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 23312	. . . 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 1087 = wceq 1542 ∈ wcel 2114 ∃*wmo 2538 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 𝒫 cpw 4556 {csn 4582 ∪ cuni 4865 ‘cfv 6500 (class class class)co 7368 ficfi 9325 fBascfbas 21312 filGencfg 21313 Topctop 22852 TopOnctopon 22869 neicnei 23056 Hauscha 23267 Filcfil 23804 fLim cflim 23893

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1o 8407 df-2o 8408 df-en 8896 df-fin 8899 df-fi 9326 df-fbas 21321 df-fg 21322 df-top 22853 df-topon 22870 df-nei 23057 df-haus 23274 df-fil 23805 df-flim 23898

This theorem is referenced by: (None)

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