Metamath Proof Explorer


Theorem onsucsuccmp

Description: The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015)

Ref Expression
Assertion onsucsuccmp ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp )

Proof

Step Hyp Ref Expression
1 suceq ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc 𝐴 = suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) )
2 suceq ( suc 𝐴 = suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc suc 𝐴 = suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) )
3 1 2 syl ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → suc suc 𝐴 = suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) )
4 3 eleq1d ( 𝐴 = if ( 𝐴 ∈ On , 𝐴 , ∅ ) → ( suc suc 𝐴 ∈ Comp ↔ suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ Comp ) )
5 0elon ∅ ∈ On
6 5 elimel if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ On
7 6 onsucsuccmpi suc suc if ( 𝐴 ∈ On , 𝐴 , ∅ ) ∈ Comp
8 4 7 dedth ( 𝐴 ∈ On → suc suc 𝐴 ∈ Comp )