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Theorem basdif0 21666
 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 4079 . . . 4 𝐵 ⊆ (𝐵 ∪ {∅})
2 undif1 4375 . . . 4 ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
31, 2sseqtrri 3931 . . 3 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4 snex 5304 . . . 4 {∅} ∈ V
5 unexg 7476 . . . 4 (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
64, 5mpan2 690 . . 3 ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7 ssexg 5197 . . 3 ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
83, 6, 7sylancr 590 . 2 ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9 elex 3428 . 2 (𝐵 ∈ TopBases → 𝐵 ∈ V)
10 indif1 4178 . . . . . . . . . . 11 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
1110unieqi 4814 . . . . . . . . . 10 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
12 unidif0 5232 . . . . . . . . . 10 ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅}) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1311, 12eqtri 2781 . . . . . . . . 9 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1413sseq2i 3923 . . . . . . . 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 4136 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑦
17 elinel2 4103 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
18 elsni 4542 . . . . . . . . . . . 12 (𝑦 ∈ {∅} → 𝑦 = ∅)
1917, 18syl 17 . . . . . . . . . . 11 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
20 0ss 4295 . . . . . . . . . . 11 ∅ ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
2119, 20eqsstrdi 3948 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 (𝐵 ∩ 𝒫 (𝑥𝑦)))
2216, 21sstrid 3905 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2322rgen 3080 . . . . . . . 8 𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
24 ralunb 4098 . . . . . . . 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 4378 . . . . . . . 8 ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
2726raleqi 3327 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2815, 25, 273bitr2i 302 . . . . . 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 4135 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
31 elinel2 4103 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
32 elsni 4542 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
3331, 32syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
3433, 20eqsstrdi 3948 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3530, 34sstrid 3905 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3635ralrimivw 3114 . . . . . . 7 (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3736rgen 3080 . . . . . 6 𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
38 ralunb 4098 . . . . . 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 3327 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4129, 39, 403bitr2i 302 . . . 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 5201 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
44 isbasisg 21660 . . . 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 21660 . . 3 (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
4742, 45, 463bitr4d 314 . 2 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
488, 9, 47pm5.21nii 383 1 ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801  TopBasesctb 21658 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-uni 4802  df-bases 21659 This theorem is referenced by: (None)
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