Metamath Proof Explorer

Theorem sn0cld

Description: The closed sets of the topology { (/) } . (Contributed by FL, 5-Jan-2009)

Ref Expression
Assertion sn0cld 
|- ( Clsd ` { (/) } ) = { (/) }

Proof

Step Hyp Ref Expression
1 0ex 
 |-  (/) e. _V
2 discld 
 |-  ( (/) e. _V -> ( Clsd ` ~P (/) ) = ~P (/) )
3 1 2 ax-mp 
 |-  ( Clsd ` ~P (/) ) = ~P (/)
4 pw0 
 |-  ~P (/) = { (/) }
5 4 fveq2i 
 |-  ( Clsd ` ~P (/) ) = ( Clsd ` { (/) } )
6 3 5 4 3eqtr3i 
 |-  ( Clsd ` { (/) } ) = { (/) }