Theorem basdif0 22840

Description: A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)

Assertion

Ref	Expression
basdif0	⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Proof of Theorem basdif0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 4141	. . . 4 ⊢ 𝐵 ⊆ (𝐵 ∪ {∅})
2		undif1 4439	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
3	1, 2	sseqtrri 3996	. . 3 ⊢ 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4		snex 5391	. . . 4 ⊢ {∅} ∈ V
5		unexg 7719	. . . 4 ⊢ (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
6	4, 5	mpan2 691	. . 3 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7		ssexg 5278	. . 3 ⊢ ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
8	3, 6, 7	sylancr 587	. 2 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9		elex 3468	. 2 ⊢ (𝐵 ∈ TopBases → 𝐵 ∈ V)
10		indif1 4245	. . . . . . . . . . 11 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
11	10	unieqi 4883	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
12		unidif0 5315	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
13	11, 12	eqtri 2752	. . . . . . . . 9 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
14	13	sseq2i 3976	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
15	14	ralbii 3075	. . . . . . 7 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
16		inss2 4201	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
17		elinel2 4165	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
18		elsni 4606	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
20		0ss 4363	. . . . . . . . . . 11 ⊢ ∅ ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
21	19, 20	eqsstrdi 3991	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
22	16, 21	sstrid 3958	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
23	22	rgen 3046	. . . . . . . 8 ⊢ ∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
24		ralunb 4160	. . . . . . . 8 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∧ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
25	23, 24	mpbiran 709	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
26		inundif 4442	. . . . . . . 8 ⊢ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
27	26	raleqi 3297	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
28	15, 25, 27	3bitr2i 299	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
29	28	ralbii 3075	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
30		inss1 4200	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
31		elinel2 4165	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
32		elsni 4606	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
34	33, 20	eqsstrdi 3991	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
35	30, 34	sstrid 3958	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
36	35	ralrimivw 3129	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
37	36	rgen 3046	. . . . . 6 ⊢ ∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
38		ralunb 4160	. . . . . 6 ⊢ (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∧ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
39	37, 38	mpbiran 709	. . . . 5 ⊢ (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
40	26	raleqi 3297	. . . . 5 ⊢ (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
41	29, 39, 40	3bitr2i 299	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
42	41	a1i 11	. . 3 ⊢ (𝐵 ∈ V → (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
43		difexg 5284	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
44		isbasisg 22834	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
45	43, 44	syl 17	. . 3 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
46		isbasisg 22834	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
47	42, 45, 46	3bitr4d 311	. 2 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
48	8, 9, 47	pm5.21nii 378	1 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  TopBasesctb 22832

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-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-sn 4590  df-pr 4592  df-uni 4872  df-bases 22833

This theorem is referenced by: (None)