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Theorem tgdif0 20707
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 3847 . . . . . . 7 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
21unieqi 4411 . . . . . 6 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3 unidif0 4798 . . . . . 6 ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = (𝐵 ∩ 𝒫 𝑥)
42, 3eqtri 2643 . . . . 5 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥)
54sseq2i 3609 . . . 4 (𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥))
65abbii 2736 . . 3 {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}
7 difexg 4768 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8 tgval 20670 . . . 4 ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
97, 8syl 17 . . 3 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10 tgval 20670 . . 3 (𝐵 ∈ V → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
116, 9, 103eqtr4a 2681 . 2 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12 difsnexi 6919 . . . . 5 ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
1312con3i 150 . . . 4 𝐵 ∈ V → ¬ (𝐵 ∖ {∅}) ∈ V)
14 fvprc 6142 . . . 4 (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
1513, 14syl 17 . . 3 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
16 fvprc 6142 . . 3 𝐵 ∈ V → (topGen‘𝐵) = ∅)
1715, 16eqtr4d 2658 . 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 1480  wcel 1987  {cab 2607  Vcvv 3186  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  cfv 5847  topGenctg 16019
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-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-topgen 16025
This theorem is referenced by:  prdsxmslem2  22244
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