Theorem relss

Description: Subclass theorem for relation predicate. Theorem 2 of Suppes p. 58. (Contributed by NM, 15-Aug-1994)

	Ref	Expression
Assertion	relss	⊢ ( 𝐴 ⊆ 𝐵 → ( Rel 𝐵 → Rel 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sstr2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ⊆ ( V × V ) → 𝐴 ⊆ ( V × V ) ) )
2		df-rel	⊢ ( Rel 𝐵 ↔ 𝐵 ⊆ ( V × V ) )
3		df-rel	⊢ ( Rel 𝐴 ↔ 𝐴 ⊆ ( V × V ) )
4	1 2 3	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( Rel 𝐵 → Rel 𝐴 ) )