MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  basdif0 Structured version   Visualization version   GIF version

Theorem basdif0 20951
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 3911 . . . 4 𝐵 ⊆ (𝐵 ∪ {∅})
2 undif1 4179 . . . 4 ((𝐵 ∖ {∅}) ∪ {∅}) = (𝐵 ∪ {∅})
31, 2sseqtr4i 3771 . . 3 𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅})
4 snex 5049 . . . 4 {∅} ∈ V
5 unexg 7116 . . . 4 (((𝐵 ∖ {∅}) ∈ TopBases ∧ {∅} ∈ V) → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
64, 5mpan2 709 . . 3 ((𝐵 ∖ {∅}) ∈ TopBases → ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V)
7 ssexg 4948 . . 3 ((𝐵 ⊆ ((𝐵 ∖ {∅}) ∪ {∅}) ∧ ((𝐵 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐵 ∈ V)
83, 6, 7sylancr 698 . 2 ((𝐵 ∖ {∅}) ∈ TopBases → 𝐵 ∈ V)
9 elex 3344 . 2 (𝐵 ∈ TopBases → 𝐵 ∈ V)
10 indif1 4006 . . . . . . . . . . 11 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
1110unieqi 4589 . . . . . . . . . 10 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅})
12 unidif0 4979 . . . . . . . . . 10 ((𝐵 ∩ 𝒫 (𝑥𝑦)) ∖ {∅}) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1311, 12eqtri 2774 . . . . . . . . 9 ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦))
1413sseq2i 3763 . . . . . . . 8 ((𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
1514ralbii 3110 . . . . . . 7 (∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
16 inss2 3969 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑦
17 inss2 3969 . . . . . . . . . . . . 13 (𝐵 ∩ {∅}) ⊆ {∅}
1817sseli 3732 . . . . . . . . . . . 12 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 ∈ {∅})
19 elsni 4330 . . . . . . . . . . . 12 (𝑦 ∈ {∅} → 𝑦 = ∅)
2018, 19syl 17 . . . . . . . . . . 11 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 = ∅)
21 0ss 4107 . . . . . . . . . . 11 ∅ ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
2220, 21syl6eqss 3788 . . . . . . . . . 10 (𝑦 ∈ (𝐵 ∩ {∅}) → 𝑦 (𝐵 ∩ 𝒫 (𝑥𝑦)))
2316, 22syl5ss 3747 . . . . . . . . 9 (𝑦 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
2423rgen 3052 . . . . . . . 8 𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
25 ralunb 3929 . . . . . . . 8 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (∀𝑦 ∈ (𝐵 ∩ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ∧ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2624, 25mpbiran 991 . . . . . . 7 (∀𝑦 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
27 inundif 4182 . . . . . . . 8 ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅})) = 𝐵
2827raleqi 3273 . . . . . . 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 3110 . . . . 5 (∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦 ∈ (𝐵 ∖ {∅})(𝑥𝑦) ⊆ ((𝐵 ∖ {∅}) ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
31 inss1 3968 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
3217sseli 3732 . . . . . . . . . . 11 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 ∈ {∅})
33 elsni 4330 . . . . . . . . . . 11 (𝑥 ∈ {∅} → 𝑥 = ∅)
3432, 33syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 = ∅)
3534, 21syl6eqss 3788 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∩ {∅}) → 𝑥 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3631, 35syl5ss 3747 . . . . . . . 8 (𝑥 ∈ (𝐵 ∩ {∅}) → (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3736ralrimivw 3097 . . . . . . 7 (𝑥 ∈ (𝐵 ∩ {∅}) → ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3837rgen 3052 . . . . . 6 𝑥 ∈ (𝐵 ∩ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))
39 ralunb 3929 . . . . . 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 991 . . . . 5 (∀𝑥 ∈ ((𝐵 ∩ {∅}) ∪ (𝐵 ∖ {∅}))∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥 ∈ (𝐵 ∖ {∅})∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
4127raleqi 3273 . . . . 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 4952 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
45 isbasisg 20945 . . . 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 20945 . . 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 367 1 ((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  cdif 3704  cun 3705  cin 3706  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   cuni 4580  TopBasesctb 20943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-sn 4314  df-pr 4316  df-uni 4581  df-bases 20944
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator