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Metamath Proof Explorer

Theorem t1ficld

Description: In a T_1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016)

Ref Expression
Hypothesis ist0.1 
|- X = U. J
Assertion t1ficld 
|- ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> A e. ( Clsd ` J ) )

Proof

Step Hyp Ref Expression
1 ist0.1 
 |-  X = U. J
2 iunid 
 |-  U_ x e. A { x } = A
3 1 ist1 
 |-  ( J e. Fre <-> ( J e. Top /\ A. x e. X { x } e. ( Clsd ` J ) ) )
4 3 simplbi 
 |-  ( J e. Fre -> J e. Top )
5 4 3ad2ant1 
 |-  ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> J e. Top )
6 simp3 
 |-  ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> A e. Fin )
7 3 simprbi 
 |-  ( J e. Fre -> A. x e. X { x } e. ( Clsd ` J ) )
8 ssralv 
 |-  ( A C_ X -> ( A. x e. X { x } e. ( Clsd ` J ) -> A. x e. A { x } e. ( Clsd ` J ) ) )
9 7 8 mpan9 
 |-  ( ( J e. Fre /\ A C_ X ) -> A. x e. A { x } e. ( Clsd ` J ) )
10 9 3adant3 
 |-  ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> A. x e. A { x } e. ( Clsd ` J ) )
11 1 iuncld 
 |-  ( ( J e. Top /\ A e. Fin /\ A. x e. A { x } e. ( Clsd ` J ) ) -> U_ x e. A { x } e. ( Clsd ` J ) )
12 5 6 10 11 syl3anc 
 |-  ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> U_ x e. A { x } e. ( Clsd ` J ) )
13 2 12 eqeltrrid 
 |-  ( ( J e. Fre /\ A C_ X /\ A e. Fin ) -> A e. ( Clsd ` J ) )