Theorem relssi

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

	Ref	Expression
Hypotheses	relssi.1	$$⊢\mathrm{Rel}A$$
	relssi.2	$$⊢⟨x,y⟩\in A\to ⟨x,y⟩\in B$$
Assertion	relssi	$$⊢A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		relssi.1	$$⊢\mathrm{Rel}A$$
2		relssi.2	$$⊢⟨x,y⟩\in A\to ⟨x,y⟩\in B$$
3		ssrel	$$⊢\mathrm{Rel}A\to \left(A\subseteq B\leftrightarrow \forall x\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)\right)$$
4	1 3	ax-mp	$$⊢A\subseteq B\leftrightarrow \forall x\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)$$
5	2	ax-gen	$$⊢\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)$$
6	4 5	mpgbir	$$⊢A\subseteq B$$