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Theorem tgdif0 21167
 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 4101 . . . . . . 7 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
21unieqi 4667 . . . . . 6 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3 unidif0 5060 . . . . . 6 ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = (𝐵 ∩ 𝒫 𝑥)
42, 3eqtri 2849 . . . . 5 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥)
54sseq2i 3855 . . . 4 (𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥))
65abbii 2944 . . 3 {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)}
7 difexg 5033 . . . 4 (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8 tgval 21130 . . . 4 ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
97, 8syl 17 . . 3 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥𝑥 ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10 tgval 21130 . . 3 (𝐵 ∈ V → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
116, 9, 103eqtr4a 2887 . 2 (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12 difsnexi 7230 . . . . 5 ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
1312con3i 152 . . . 4 𝐵 ∈ V → ¬ (𝐵 ∖ {∅}) ∈ V)
14 fvprc 6426 . . . 4 (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
1513, 14syl 17 . . 3 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
16 fvprc 6426 . . 3 𝐵 ∈ V → (topGen‘𝐵) = ∅)
1715, 16eqtr4d 2864 . 2 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
1811, 17pm2.61i 177 1 (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1656   ∈ wcel 2164  {cab 2811  Vcvv 3414   ∖ cdif 3795   ∩ cin 3797   ⊆ wss 3798  ∅c0 4144  𝒫 cpw 4378  {csn 4397  ∪ cuni 4658  ‘cfv 6123  topGenctg 16451 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-topgen 16457 This theorem is referenced by:  prdsxmslem2  22704
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