Theorem basdif0 20568

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 3738	. . . 4 ⊢ 𝐵 ⊆ (𝐵 ∪ {∅})
2		undif1 3995	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
3	1, 2	sseqtr4i 3601	. . 3 ⊢ 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4		snex 4835	. . . 4 ⊢ {∅} ∈ V
5		unexg 6857	. . . 4 ⊢ (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
6	4, 5	mpan2 703	. . 3 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7		ssexg 4732	. . 3 ⊢ ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
8	3, 6, 7	sylancr 694	. 2 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9		elex 3185	. 2 ⊢ (𝐵 ∈ TopBases → 𝐵 ∈ V)
10		indif1 3830	. . . . . . . . . . 11 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
11	10	unieqi 4381	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
12		unidif0 4764	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
13	11, 12	eqtri 2632	. . . . . . . . 9 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
14	13	sseq2i 3593	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
15	14	ralbii 2963	. . . . . . 7 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
16		inss2 3796	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
17		inss2 3796	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ {∅}) ⊆ {∅}
18	17	sseli 3564	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
19		elsni 4142	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
20	18, 19	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
21		0ss 3924	. . . . . . . . . . 11 ⊢ ∅ ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
22	20, 21	syl6eqss 3618	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
23	16, 22	syl5ss 3579	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
24	23	rgen 2906	. . . . . . . 8 ⊢ ∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
25		ralunb 3756	. . . . . . . 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))))
26	24, 25	mpbiran 955	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
27		inundif 3998	. . . . . . . 8 ⊢ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
28	27	raleqi 3119	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
29	15, 26, 28	3bitr2i 287	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
30	29	ralbii 2963	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
31		inss1 3795	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
32	17	sseli 3564	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
33		elsni 4142	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
35	34, 21	syl6eqss 3618	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
36	31, 35	syl5ss 3579	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
37	36	ralrimivw 2950	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
38	37	rgen 2906	. . . . . 6 ⊢ ∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
39		ralunb 3756	. . . . . 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))))
40	38, 39	mpbiran 955	. . . . 5 ⊢ (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
41	27	raleqi 3119	. . . . 5 ⊢ (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
42	30, 40, 41	3bitr2i 287	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
43	42	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))))
44		difexg 4735	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
45		isbasisg 20562	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
46	44, 45	syl 17	. . 3 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
47		isbasisg 20562	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
48	43, 46, 47	3bitr4d 299	. 2 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
49	8, 9, 48	pm5.21nii 367	1 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  TopBasesctb 20520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-bases 20522

This theorem is referenced by: (None)