qtopbasss - Intuitionistic Logic Explorer

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Theorem qtopbasss 14261

Description: The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)

**Hypotheses**
Ref	Expression
qtopbas.1	⊢ 𝑆 ⊆ ℝ^*
qtopbas.max	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → sup({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆)
qtopbas.min	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → inf({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆)

**Assertion**
Ref	Expression
qtopbasss	⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Distinct variable group: 𝑥,𝑦,𝑆

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

Step	Hyp	Ref	Expression
1		iooex 9920	. . 3 ⊢ (,) ∈ V
2	1	imaex 4995	. 2 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ V
3		qtopbas.1	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℝ^*
4	3	sseli 3163	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ℝ^*)
5	3	sseli 3163	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℝ^*)
6	4, 5	anim12i 338	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
7	3	sseli 3163	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ^*)
8	3	sseli 3163	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ ℝ^*)
9	7, 8	anim12i 338	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝑣 ∈ ℝ^* ∧ 𝑢 ∈ ℝ^*))
10		iooinsup 11298	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝑣 ∈ ℝ^ ∧ 𝑢 ∈ ℝ^)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) = (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^*, < )))
11	6, 9, 10	syl2an 289	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) = (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^, < )))
12		qtopbas.max	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → sup({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆)
13	12	rgen2a 2541	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 sup({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆
14		preq12 3683	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑧) → {𝑥, 𝑦} = {𝑣, 𝑧})
15		prcom 3680	. . . . . . . . . . . . . 14 ⊢ {𝑣, 𝑧} = {𝑧, 𝑣}
16	14, 15	eqtrdi 2236	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑧) → {𝑥, 𝑦} = {𝑧, 𝑣})
17	16	supeq1d 6999	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑧) → sup({𝑥, 𝑦}, ℝ^, < ) = sup({𝑧, 𝑣}, ℝ^, < ))
18	17	eleq1d 2256	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑧) → (sup({𝑥, 𝑦}, ℝ^, < ) ∈ 𝑆 ↔ sup({𝑧, 𝑣}, ℝ^, < ) ∈ 𝑆))
19	18	rspc2gv 2865	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 sup({𝑥, 𝑦}, ℝ^, < ) ∈ 𝑆 → sup({𝑧, 𝑣}, ℝ^, < ) ∈ 𝑆))
20	13, 19	mpi 15	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → sup({𝑧, 𝑣}, ℝ^*, < ) ∈ 𝑆)
21	20	ancoms 268	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → sup({𝑧, 𝑣}, ℝ^*, < ) ∈ 𝑆)
22		qtopbas.min	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → inf({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆)
23	22	rgen2a 2541	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 inf({𝑥, 𝑦}, ℝ^*, < ) ∈ 𝑆
24		preq12 3683	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → {𝑥, 𝑦} = {𝑤, 𝑢})
25	24	infeq1d 7024	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → inf({𝑥, 𝑦}, ℝ^, < ) = inf({𝑤, 𝑢}, ℝ^, < ))
26	25	eleq1d 2256	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) → (inf({𝑥, 𝑦}, ℝ^, < ) ∈ 𝑆 ↔ inf({𝑤, 𝑢}, ℝ^, < ) ∈ 𝑆))
27	26	rspc2gv 2865	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 inf({𝑥, 𝑦}, ℝ^, < ) ∈ 𝑆 → inf({𝑤, 𝑢}, ℝ^, < ) ∈ 𝑆))
28	23, 27	mpi 15	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → inf({𝑤, 𝑢}, ℝ^*, < ) ∈ 𝑆)
29		df-ov 5891	. . . . . . . . 9 ⊢ (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^, < )) = ((,)‘⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩)
30		opelxpi 4670	. . . . . . . . . 10 ⊢ ((sup({𝑧, 𝑣}, ℝ^, < ) ∈ 𝑆 ∧ inf({𝑤, 𝑢}, ℝ^, < ) ∈ 𝑆) → ⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩ ∈ (𝑆 × 𝑆))
31		ioof 9984	. . . . . . . . . . . 12 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
32		ffun 5380	. . . . . . . . . . . 12 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
33	31, 32	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun (,)
34		xpss12 4745	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℝ^* ∧ 𝑆 ⊆ ℝ^) → (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*))
35	3, 3, 34	mp2an 426	. . . . . . . . . . . 12 ⊢ (𝑆 × 𝑆) ⊆ (ℝ^* × ℝ^*)
36	31	fdmi 5385	. . . . . . . . . . . 12 ⊢ dom (,) = (ℝ^* × ℝ^*)
37	35, 36	sseqtrri 3202	. . . . . . . . . . 11 ⊢ (𝑆 × 𝑆) ⊆ dom (,)
38		funfvima2 5762	. . . . . . . . . . 11 ⊢ ((Fun (,) ∧ (𝑆 × 𝑆) ⊆ dom (,)) → (⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩ ∈ (𝑆 × 𝑆) → ((,)‘⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩) ∈ ((,) “ (𝑆 × 𝑆))))
39	33, 37, 38	mp2an 426	. . . . . . . . . 10 ⊢ (⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩ ∈ (𝑆 × 𝑆) → ((,)‘⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩) ∈ ((,) “ (𝑆 × 𝑆)))
40	30, 39	syl 14	. . . . . . . . 9 ⊢ ((sup({𝑧, 𝑣}, ℝ^, < ) ∈ 𝑆 ∧ inf({𝑤, 𝑢}, ℝ^, < ) ∈ 𝑆) → ((,)‘⟨sup({𝑧, 𝑣}, ℝ^, < ), inf({𝑤, 𝑢}, ℝ^, < )⟩) ∈ ((,) “ (𝑆 × 𝑆)))
41	29, 40	eqeltrid 2274	. . . . . . . 8 ⊢ ((sup({𝑧, 𝑣}, ℝ^, < ) ∈ 𝑆 ∧ inf({𝑤, 𝑢}, ℝ^, < ) ∈ 𝑆) → (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^, < )) ∈ ((,) “ (𝑆 × 𝑆)))
42	21, 28, 41	syl2an 289	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑤 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^, < )) ∈ ((,) “ (𝑆 × 𝑆)))
43	42	an4s 588	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (sup({𝑧, 𝑣}, ℝ^, < )(,)inf({𝑤, 𝑢}, ℝ^, < )) ∈ ((,) “ (𝑆 × 𝑆)))
44	11, 43	eqeltrd 2264	. . . . 5 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
45	44	ralrimivva 2569	. . . 4 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
46	45	rgen2a 2541	. . 3 ⊢ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))
47		ffn 5377	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
48	31, 47	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
49		ineq1 3341	. . . . . . . 8 ⊢ (𝑥 = ((,)‘𝑡) → (𝑥 ∩ 𝑦) = (((,)‘𝑡) ∩ 𝑦))
50	49	eleq1d 2256	. . . . . . 7 ⊢ (𝑥 = ((,)‘𝑡) → ((𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ (((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
51	50	ralbidv 2487	. . . . . 6 ⊢ (𝑥 = ((,)‘𝑡) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
52	51	ralima 5769	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*)) → (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
53	48, 35, 52	mp2an 426	. . . 4 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)))
54		fveq2 5527	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = ((,)‘⟨𝑧, 𝑤⟩))
55		df-ov 5891	. . . . . . . . . 10 ⊢ (𝑧(,)𝑤) = ((,)‘⟨𝑧, 𝑤⟩)
56	54, 55	eqtr4di 2238	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((,)‘𝑡) = (𝑧(,)𝑤))
57	56	ineq1d 3347	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (((,)‘𝑡) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ 𝑦))
58	57	eleq1d 2256	. . . . . . 7 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → ((((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
59	58	ralbidv 2487	. . . . . 6 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))))
60		ineq2 3342	. . . . . . . . . 10 ⊢ (𝑦 = ((,)‘𝑡) → ((𝑧(,)𝑤) ∩ 𝑦) = ((𝑧(,)𝑤) ∩ ((,)‘𝑡)))
61	60	eleq1d 2256	. . . . . . . . 9 ⊢ (𝑦 = ((,)‘𝑡) → (((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
62	61	ralima 5769	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ (𝑆 × 𝑆) ⊆ (ℝ^ × ℝ^*)) → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆))))
63	48, 35, 62	mp2an 426	. . . . . . 7 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)))
64		fveq2 5527	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = ((,)‘⟨𝑣, 𝑢⟩))
65		df-ov 5891	. . . . . . . . . . 11 ⊢ (𝑣(,)𝑢) = ((,)‘⟨𝑣, 𝑢⟩)
66	64, 65	eqtr4di 2238	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((,)‘𝑡) = (𝑣(,)𝑢))
67	66	ineq2d 3348	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → ((𝑧(,)𝑤) ∩ ((,)‘𝑡)) = ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)))
68	67	eleq1d 2256	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑣, 𝑢⟩ → (((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
69	68	ralxp 4782	. . . . . . 7 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)((𝑧(,)𝑤) ∩ ((,)‘𝑡)) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
70	63, 69	bitri 184	. . . . . 6 ⊢ (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))((𝑧(,)𝑤) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
71	59, 70	bitrdi 196	. . . . 5 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆))))
72	71	ralxp 4782	. . . 4 ⊢ (∀𝑡 ∈ (𝑆 × 𝑆)∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(((,)‘𝑡) ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
73	53, 72	bitri 184	. . 3 ⊢ (∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆)) ↔ ∀𝑧 ∈ 𝑆 ∀𝑤 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ∀𝑢 ∈ 𝑆 ((𝑧(,)𝑤) ∩ (𝑣(,)𝑢)) ∈ ((,) “ (𝑆 × 𝑆)))
74	46, 73	mpbir 146	. 2 ⊢ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))
75		fiinbas 13789	. 2 ⊢ ((((,) “ (𝑆 × 𝑆)) ∈ V ∧ ∀𝑥 ∈ ((,) “ (𝑆 × 𝑆))∀𝑦 ∈ ((,) “ (𝑆 × 𝑆))(𝑥 ∩ 𝑦) ∈ ((,) “ (𝑆 × 𝑆))) → ((,) “ (𝑆 × 𝑆)) ∈ TopBases)
76	2, 74, 75	mp2an 426	1 ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ∀wral 2465 Vcvv 2749 ∩ cin 3140 ⊆ wss 3141 𝒫 cpw 3587 {cpr 3605 ⟨cop 3607 × cxp 4636 dom cdm 4638 “ cima 4641 Fun wfun 5222 Fn wfn 5223 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 supcsup 6994 infcinf 6995 ℝcr 7823 ℝ^*cxr 8004 < clt 8005 (,)cioo 9901 TopBasesctb 13782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-rp 9667 df-xneg 9785 df-ioo 9905 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-bases 13783

This theorem is referenced by: qtopbas 14262 retopbas 14263

W3C validator