Theorem relssi

Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998)

	Ref	Expression
Hypotheses	relssi.1	⊢ Rel 𝐴
	relssi.2	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
Assertion	relssi	⊢ 𝐴 ⊆ 𝐵

Proof

Step	Hyp	Ref	Expression
1		relssi.1	⊢ Rel 𝐴
2		relssi.2	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
3		ssrel	⊢ ( Rel 𝐴 → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
4	1 3	ax-mp	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
5	2	ax-gen	⊢ ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
6	4 5	mpgbir	⊢ 𝐴 ⊆ 𝐵