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Theorem tgdif0 20844
Description: A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)
Assertion
Ref Expression
tgdif0 (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)

Proof of Theorem tgdif0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 indif1 3904 . . . . . . 7 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
21unieqi 4477 . . . . . 6 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3 unidif0 4868 . . . . . 6 ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = (𝐵 ∩ 𝒫 𝑥)
42, 3eqtri 2673 . . . . 5 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥)
54sseq2i 3663 . . . 4 (𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥))
65abbii 2768 . . 3 {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}
7 difexg 4841 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8 tgval 20807 . . . 4 ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
97, 8syl 17 . . 3 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10 tgval 20807 . . 3 (𝐵 ∈ V → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
116, 9, 103eqtr4a 2711 . 2 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12 difsnexi 7012 . . . . 5 ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
1312con3i 150 . . . 4 𝐵 ∈ V → ¬ (𝐵 ∖ {∅}) ∈ V)
14 fvprc 6223 . . . 4 (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
1513, 14syl 17 . . 3 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
16 fvprc 6223 . . 3 𝐵 ∈ V → (topGen‘𝐵) = ∅)
1715, 16eqtr4d 2688 . 2 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
1811, 17pm2.61i 176 1 (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cfv 5926  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151
This theorem is referenced by:  prdsxmslem2  22381
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