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Theorem basdif0 20668
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 3754 . . . 4 𝐵 ⊆ (𝐵 ∪ {∅})
2 undif1 4015 . . . 4 ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
31, 2sseqtr4i 3617 . . 3 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4 snex 4869 . . . 4 {∅} ∈ V
5 unexg 6912 . . . 4 (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
64, 5mpan2 706 . . 3 ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7 ssexg 4764 . . 3 ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
83, 6, 7sylancr 694 . 2 ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9 elex 3198 . 2 (𝐵 ∈ TopBases → 𝐵 ∈ V)
10 indif1 3847 . . . . . . . . . . 11 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
1110unieqi 4411 . . . . . . . . . 10 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
12 unidif0 4798 . . . . . . . . . 10 ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅}) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1311, 12eqtri 2643 . . . . . . . . 9 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1413sseq2i 3609 . . . . . . . 8 ((𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
1514ralbii 2974 . . . . . . 7 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
16 inss2 3812 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑦
17 inss2 3812 . . . . . . . . . . . . 13 (𝐵 ∩ {∅}) ⊆ {∅}
1817sseli 3579 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
19 elsni 4165 . . . . . . . . . . . 12 (𝑦 ∈ {∅} → 𝑦 = ∅)
2018, 19syl 17 . . . . . . . . . . 11 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
21 0ss 3944 . . . . . . . . . . 11 ∅ ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
2220, 21syl6eqss 3634 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 (𝐵 ∩ 𝒫 (𝑥𝑦)))
2316, 22syl5ss 3594 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2423rgen 2917 . . . . . . . 8 𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
25 ralunb 3772 . . . . . . . 8 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2624, 25mpbiran 952 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
27 inundif 4018 . . . . . . . 8 ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
2827raleqi 3131 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2915, 26, 283bitr2i 288 . . . . . 6 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3029ralbii 2974 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
31 inss1 3811 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
3217sseli 3579 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
33 elsni 4165 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
3432, 33syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
3534, 21syl6eqss 3634 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3631, 35syl5ss 3594 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3736ralrimivw 2961 . . . . . . 7 (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3837rgen 2917 . . . . . 6 𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
39 ralunb 3772 . . . . . 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 952 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4127raleqi 3131 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4230, 40, 413bitr2i 288 . . . 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 4768 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
45 isbasisg 20662 . . . 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 20662 . . 3 (𝐵 ∈ V → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
4843, 46, 473bitr4d 300 . 2 (𝐵 ∈ V → ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases))
498, 9, 48pm5.21nii 368 1 ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  TopBasesctb 20620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151  df-uni 4403  df-bases 20622
This theorem is referenced by: (None)
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