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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.
StepHypRef Expression
1 ssun1 3738 . . . 4 𝐵 ⊆ (𝐵 ∪ {∅})
2 undif1 3995 . . . 4 ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
31, 2sseqtr4i 3601 . . 3 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4 snex 4835 . . . 4 {∅} ∈ V
5 unexg 6857 . . . 4 (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
64, 5mpan2 703 . . 3 ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7 ssexg 4732 . . 3 ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
83, 6, 7sylancr 694 . 2 ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9 elex 3185 . 2 (𝐵 ∈ TopBases → 𝐵 ∈ V)
10 indif1 3830 . . . . . . . . . . 11 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
1110unieqi 4381 . . . . . . . . . 10 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
12 unidif0 4764 . . . . . . . . . 10 ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅}) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1311, 12eqtri 2632 . . . . . . . . 9 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1413sseq2i 3593 . . . . . . . 8 ((𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
1514ralbii 2963 . . . . . . 7 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
16 inss2 3796 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑦
17 inss2 3796 . . . . . . . . . . . . 13 (𝐵 ∩ {∅}) ⊆ {∅}
1817sseli 3564 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
19 elsni 4142 . . . . . . . . . . . 12 (𝑦 ∈ {∅} → 𝑦 = ∅)
2018, 19syl 17 . . . . . . . . . . 11 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
21 0ss 3924 . . . . . . . . . . 11 ∅ ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
2220, 21syl6eqss 3618 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 (𝐵 ∩ 𝒫 (𝑥𝑦)))
2316, 22syl5ss 3579 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2423rgen 2906 . . . . . . . 8 𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
25 ralunb 3756 . . . . . . . 8 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2624, 25mpbiran 955 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
27 inundif 3998 . . . . . . . 8 ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
2827raleqi 3119 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2915, 26, 283bitr2i 287 . . . . . 6 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3029ralbii 2963 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
31 inss1 3795 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
3217sseli 3564 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
33 elsni 4142 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
3432, 33syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
3534, 21syl6eqss 3618 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3631, 35syl5ss 3579 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3736ralrimivw 2950 . . . . . . 7 (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3837rgen 2906 . . . . . 6 𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
39 ralunb 3756 . . . . . 6 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
4038, 39mpbiran 955 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4127raleqi 3119 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4230, 40, 413bitr2i 287 . . . 4 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4342a1i 11 . . 3 (𝐵 ∈ V → (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
44 difexg 4735 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
45 isbasisg 20562 . . . 4 ((𝐵 ∖ {∅}) ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦))))
4644, 45syl 17 . . 3 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦))))
47 isbasisg 20562 . . 3 (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
4843, 46, 473bitr4d 299 . 2 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
498, 9, 48pm5.21nii 367 1 ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
 Colors of variables: wff setvar class 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)
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