Theorem tgdif0 23117

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 4243	. . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
2	1	unieqi 4888	. . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3		unidif0 5331	. . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥)
4	2, 3	eqtri 2792	. . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥)
5	4	sseq2i 3974	. . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))
6	5	abbii 2836	. . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}
7		difexg 5300	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8		tgval 23080	. . . 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 23080	. . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
11	6, 9, 10	3eqtr4a 2830	. 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12		difsnexi 7759	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
13		fvprc 6874	. . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
14	12, 13	nsyl5 160	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
15		fvprc 6874	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅)
16	14, 15	eqtr4d 2807	. 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
17	11, 16	pm2.61i 184	1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  {cab 2747  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876  ‘cfv 6537  topGenctg 17489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-topgen 17495

This theorem is referenced by:  prdsxmslem2  24654