Theorem tgdif0

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)

	Ref	Expression
Assertion	tgdif0	⊢ ( topGen ‘ ( 𝐵 ∖ { ∅ } ) ) = ( topGen ‘ 𝐵 )

Proof

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