Theorem basdif0 23071

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 4133	. . . 4 ⊢ 𝐵 ⊆ (𝐵 ∪ {∅})
2		undif1 4433	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
3	1, 2	sseqtrri 3988	. . 3 ⊢ 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4		snex 5401	. . . 4 ⊢ {∅} ∈ V
5		unexg 7730	. . . 4 ⊢ (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
6	4, 5	mpan2 703	. . 3 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7		ssexg 5284	. . 3 ⊢ ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
8	3, 6, 7	sylancr 598	. 2 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9		elex 3478	. 2 ⊢ (𝐵 ∈ TopBases → 𝐵 ∈ V)
10		indif1 4237	. . . . . . . . . . 11 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
11	10	unieqi 4880	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅})
12		unidif0 5321	. . . . . . . . . 10 ⊢ ∪ ((𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
13	11, 12	eqtri 2788	. . . . . . . . 9 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
14	13	sseq2i 3968	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
15	14	ralbii 3111	. . . . . . 7 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
16		inss2 4192	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
17		elinel2 4157	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
18		elsni 4602	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
19	17, 18	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
20		0ss 4357	. . . . . . . . . . 11 ⊢ ∅ ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
21	19, 20	eqsstrdi 3983	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
22	16, 21	sstrid 3950	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
23	22	rgen 3081	. . . . . . . 8 ⊢ ∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
24		ralunb 4152	. . . . . . . 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 721	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
26		inundif 4436	. . . . . . . 8 ⊢ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
27	26	raleqi 3321	. . . . . . 7 ⊢ (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
28	15, 25, 27	3bitr2i 302	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
29	28	ralbii 3111	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
30		inss1 4191	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
31		elinel2 4157	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
32		elsni 4602	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
33	31, 32	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
34	33, 20	eqsstrdi 3983	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
35	30, 34	sstrid 3950	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
36	35	ralrimivw 3161	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
37	36	rgen 3081	. . . . . 6 ⊢ ∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))
38		ralunb 4152	. . . . . 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 721	. . . . 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 3321	. . . . 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 302	. . . 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 5290	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
44		isbasisg 23065	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
45	43, 44	syl 18	. . 3 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥 ∩ 𝑦) ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥 ∩ 𝑦))))
46		isbasisg 23065	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
47	42, 45, 46	3bitr4d 314	. 2 ⊢ (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
48	8, 9, 47	pm5.21nii 381	1 ⊢ ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {csn 4585  ∪ cuni 4868  TopBasesctb 23063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-uni 4869  df-bases 23064

This theorem is referenced by: (None)