Theorem tgdif0 23199

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 4235	. . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
2	1	unieqi 4886	. . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3		unidif0 5332	. . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥)
4	2, 3	eqtri 2788	. . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥)
5	4	sseq2i 3967	. . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))
6	5	abbii 2832	. . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}
7		difexg 5302	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8		tgval 23162	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
9	7, 8	syl 18	. . 3 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)})
10		tgval 23162	. . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
11	6, 9, 10	3eqtr4a 2826	. 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12		difsnexi 7766	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
13		fvprc 6877	. . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
14	12, 13	nsyl5 160	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
15		fvprc 6877	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅)
16	14, 15	eqtr4d 2803	. 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
17	11, 16	pm2.61i 184	1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  {cab 2743  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874  ‘cfv 6540  topGenctg 17512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-topgen 17518

This theorem is used by:  prdsxmslem2  24737