Theorem tgdif0 22879

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 4245	. . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
2	1	unieqi 4883	. . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅})
3		unidif0 5315	. . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥)
4	2, 3	eqtri 2752	. . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥)
5	4	sseq2i 3976	. . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))
6	5	abbii 2796	. . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}
7		difexg 5284	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V)
8		tgval 22842	. . . 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 22842	. . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)})
11	6, 9, 10	3eqtr4a 2790	. 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
12		difsnexi 7737	. . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V)
13		fvprc 6850	. . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
14	12, 13	nsyl5 159	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅)
15		fvprc 6850	. . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅)
16	14, 15	eqtr4d 2767	. 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵))
17	11, 16	pm2.61i 182	1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ‘cfv 6511  topGenctg 17400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-topgen 17406

This theorem is referenced by:  prdsxmslem2  24417