Metamath Proof Explorer

Theorem ssct

Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017)

Ref Expression
Assertion ssct 
|- ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om )

Proof

Step Hyp Ref Expression
1 ctex 
 |-  ( B ~<_ _om -> B e. _V )
2 ssdomg 
 |-  ( B e. _V -> ( A C_ B -> A ~<_ B ) )
3 1 2 syl 
 |-  ( B ~<_ _om -> ( A C_ B -> A ~<_ B ) )
4 3 impcom 
 |-  ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ B )
5 domtr 
 |-  ( ( A ~<_ B /\ B ~<_ _om ) -> A ~<_ _om )
6 4 5 sylancom 
 |-  ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om )