Metamath Proof Explorer


Theorem isperf3

Description: A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016)

Ref Expression
Hypothesis lpfval.1 𝑋 = 𝐽
Assertion isperf3 ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥𝑋 ¬ { 𝑥 } ∈ 𝐽 ) )

Proof

Step Hyp Ref Expression
1 lpfval.1 𝑋 = 𝐽
2 1 isperf2 ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) )
3 dfss3 ( 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥𝑋 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) )
4 1 maxlp ( 𝐽 ∈ Top → ( 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ( 𝑥𝑋 ∧ ¬ { 𝑥 } ∈ 𝐽 ) ) )
5 4 baibd ( ( 𝐽 ∈ Top ∧ 𝑥𝑋 ) → ( 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ¬ { 𝑥 } ∈ 𝐽 ) )
6 5 ralbidva ( 𝐽 ∈ Top → ( ∀ 𝑥𝑋 𝑥 ∈ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥𝑋 ¬ { 𝑥 } ∈ 𝐽 ) )
7 3 6 syl5bb ( 𝐽 ∈ Top → ( 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ↔ ∀ 𝑥𝑋 ¬ { 𝑥 } ∈ 𝐽 ) )
8 7 pm5.32i ( ( 𝐽 ∈ Top ∧ 𝑋 ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝑋 ) ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥𝑋 ¬ { 𝑥 } ∈ 𝐽 ) )
9 2 8 bitri ( 𝐽 ∈ Perf ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥𝑋 ¬ { 𝑥 } ∈ 𝐽 ) )