Theorem ssct

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

	Ref	Expression
Assertion	ssct	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω )

Proof

Step	Hyp	Ref	Expression
1		ctex	⊢ ( 𝐵 ≼ ω → 𝐵 ∈ V )
2		ssdomg	⊢ ( 𝐵 ∈ V → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )
3	1 2	syl	⊢ ( 𝐵 ≼ ω → ( 𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵 ) )
4	3	impcom	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ 𝐵 )
5		domtr	⊢ ( ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω )
6	4 5	sylancom	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω ) → 𝐴 ≼ ω )