Theorem lpdifsn

Description: P is a limit point of S iff it is a limit point of S \ { P } . (Contributed by Mario Carneiro, 25-Dec-2016)

	Ref	Expression
Hypothesis	lpfval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	lpdifsn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	⊢ 𝑋 = ∪ 𝐽
2	1	islp	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )
3		ssdifss	⊢ ( 𝑆 ⊆ 𝑋 → ( 𝑆 ∖ { 𝑃 } ) ⊆ 𝑋 )
4	1	islp	⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑆 ∖ { 𝑃 } ) ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑆 ∖ { 𝑃 } ) ∖ { 𝑃 } ) ) ) )
5	3 4	sylan2	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑆 ∖ { 𝑃 } ) ∖ { 𝑃 } ) ) ) )
6		difabs	⊢ ( ( 𝑆 ∖ { 𝑃 } ) ∖ { 𝑃 } ) = ( 𝑆 ∖ { 𝑃 } )
7	6	fveq2i	⊢ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑆 ∖ { 𝑃 } ) ∖ { 𝑃 } ) ) = ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) )
8	7	eleq2i	⊢ ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑆 ∖ { 𝑃 } ) ∖ { 𝑃 } ) ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) )
9	5 8	bitrdi	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ↔ 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )
10	2 9	bitr4d	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ↔ 𝑃 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑃 } ) ) ) )