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Theorem hausflim 23866

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 23216	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2		hausflimi 23865	. . . 4 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
3	2	ralrimivw 3125	. . 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 22802	. . . . . . . . . . . . . 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 4760	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ⊆ 𝑋)
11	9	snn0d 4727	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ≠ ∅)
12		neifil 23765	. . . . . . . . . . . . 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 23733	. . . . . . . . . . . 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 4760	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ⊆ 𝑋)
18	16	snn0d 4727	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ≠ ∅)
19		neifil 23765	. . . . . . . . . . . . 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 23733	. . . . . . . . . . . 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 23754	. . . . . . . . . . 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 22992	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑧}) ⊆ 𝒫 𝑋)
26	6	neisspw 22992	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑤}) ⊆ 𝒫 𝑋)
27	25, 26	unssd 4143	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 14 ⊢ ((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))
31		filn0 23747	. . . . . . . . . . . . . . 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 4355	. . . . . . . . . . . . . 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 22791	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
39	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑋 ∈ 𝐽)
40		fsubbas 23752	. . . . . . . . . . . . 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 23763	. . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑧}) ∈ V
48		fvex 6835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑤}) ∈ V
49	47, 48	unex 7680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V
50		ssfii 9309	. . . . . . . . . . . . . . . . . . . . 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 23757	. . . . . . . . . . . . . . . . . . . . 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 23856	. . . . . . . . . . . . . . . . . . 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 4145	. . . . . . . . . . . . . . . . . 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 23856	. . . . . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . . . . . . 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 2811	. . . . . . . . . . . . . . . . . . . 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 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 2938	. . . . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . 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 2542	. . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . 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 2926	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ (𝑢 ∩ 𝑣) = ∅)
83	82	ralbii 3075	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅)
84		ralnex 3055	. . . . . . . . . . . 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 3075	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
87		ralnex 3055	. . . . . . . . . 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 3081	. . . . . . . . 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 3172	. . 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 23238	. . . 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 2531 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3436 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 𝒫 cpw 4551 {csn 4577 ∪ cuni 4858 ‘cfv 6482 (class class class)co 7349 ficfi 9300 fBascfbas 21249 filGencfg 21250 Topctop 22778 TopOnctopon 22795 neicnei 22982 Hauscha 23193 Filcfil 23730 fLim cflim 23819

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1o 8388 df-2o 8389 df-en 8873 df-fin 8876 df-fi 9301 df-fbas 21258 df-fg 21259 df-top 22779 df-topon 22796 df-nei 22983 df-haus 23200 df-fil 23731 df-flim 23824

This theorem is referenced by: (None)

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