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Theorem basdif0 22319
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.
StepHypRef Expression
1 ssun1 4137 . . . 4 𝐵 ⊆ (𝐵 ∪ {∅})
2 undif1 4440 . . . 4 ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
31, 2sseqtrri 3986 . . 3 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4 snex 5393 . . . 4 {∅} ∈ V
5 unexg 7688 . . . 4 (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
64, 5mpan2 690 . . 3 ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7 ssexg 5285 . . 3 ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
83, 6, 7sylancr 588 . 2 ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9 elex 3466 . 2 (𝐵 ∈ TopBases → 𝐵 ∈ V)
10 indif1 4236 . . . . . . . . . . 11 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
1110unieqi 4883 . . . . . . . . . 10 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
12 unidif0 5320 . . . . . . . . . 10 ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅}) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1311, 12eqtri 2765 . . . . . . . . 9 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1413sseq2i 3978 . . . . . . . 8 ((𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
1514ralbii 3097 . . . . . . 7 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
16 inss2 4194 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑦
17 elinel2 4161 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
18 elsni 4608 . . . . . . . . . . . 12 (𝑦 ∈ {∅} → 𝑦 = ∅)
1917, 18syl 17 . . . . . . . . . . 11 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
20 0ss 4361 . . . . . . . . . . 11 ∅ ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
2119, 20eqsstrdi 4003 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 (𝐵 ∩ 𝒫 (𝑥𝑦)))
2216, 21sstrid 3960 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2322rgen 3067 . . . . . . . 8 𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
24 ralunb 4156 . . . . . . . 8 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2523, 24mpbiran 708 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
26 inundif 4443 . . . . . . . 8 ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
2726raleqi 3314 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2815, 25, 273bitr2i 299 . . . . . 6 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2928ralbii 3097 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
30 inss1 4193 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
31 elinel2 4161 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
32 elsni 4608 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
3331, 32syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
3433, 20eqsstrdi 4003 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3530, 34sstrid 3960 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3635ralrimivw 3148 . . . . . . 7 (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3736rgen 3067 . . . . . 6 𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
38 ralunb 4156 . . . . . 6 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
3937, 38mpbiran 708 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4026raleqi 3314 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4129, 39, 403bitr2i 299 . . . 4 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4241a1i 11 . . 3 (𝐵 ∈ V → (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43 difexg 5289 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
44 isbasisg 22313 . . . 4 ((𝐵 ∖ {∅}) ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦))))
4543, 44syl 17 . . 3 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦))))
46 isbasisg 22313 . . 3 (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
4742, 45, 463bitr4d 311 . 2 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
488, 9, 47pm5.21nii 380 1 ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  cdif 3912  cun 3913  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870  TopBasesctb 22311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-sn 4592  df-pr 4594  df-uni 4871  df-bases 22312
This theorem is referenced by: (None)
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