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Theorem tgdif0 22888
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 4267 . . . . . . 7 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
21unieqi 4915 . . . . . 6 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3 unidif0 5354 . . . . . 6 ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = (𝐵 ∩ 𝒫 𝑥)
42, 3eqtri 2755 . . . . 5 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥)
54sseq2i 4007 . . . 4 (𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥))
65abbii 2797 . . 3 {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}
7 difexg 5323 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8 tgval 22851 . . . 4 ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
97, 8syl 17 . . 3 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10 tgval 22851 . . 3 (𝐵 ∈ V → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
116, 9, 103eqtr4a 2793 . 2 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12 difsnexi 7757 . . . 4 ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
13 fvprc 6883 . . . 4 (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
1412, 13nsyl5 159 . . 3 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
15 fvprc 6883 . . 3 𝐵 ∈ V → (topGen‘𝐵) = ∅)
1614, 15eqtr4d 2770 . 2 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
1711, 16pm2.61i 182 1 (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  {cab 2704  Vcvv 3469  cdif 3941  cin 3943  wss 3944  c0 4318  𝒫 cpw 4598  {csn 4624   cuni 4903  cfv 6542  topGenctg 17412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17418
This theorem is referenced by:  prdsxmslem2  24431
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