Theorem tgdif0 23015

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.

Step	Hyp	Ref	Expression
1		indif1 4288	. . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
2	1	unieqi 4924	. . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3		unidif0 5366	. . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥)
4	2, 3	eqtri 2763	. . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥)
5	4	sseq2i 4025	. . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))
6	5	abbii 2807	. . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}
7		difexg 5335	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8		tgval 22978	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
9	7, 8	syl 17	. . 3 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10		tgval 22978	. . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
11	6, 9, 10	3eqtr4a 2801	. 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12		difsnexi 7780	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
13		fvprc 6899	. . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
14	12, 13	nsyl5 159	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
15		fvprc 6899	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅)
16	14, 15	eqtr4d 2778	. 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
17	11, 16	pm2.61i 182	1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ‘cfv 6563  topGenctg 17484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490

This theorem is referenced by:  prdsxmslem2  24558