Theorem relssdv

Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004)

	Ref	Expression
Hypotheses	relssdv.1	$$⊢\phi \to \mathrm{Rel}A$$
	relssdv.2	$$⊢\phi \to \left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)$$
Assertion	relssdv	$$⊢\phi \to A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		relssdv.1	$$⊢\phi \to \mathrm{Rel}A$$
2		relssdv.2	$$⊢\phi \to \left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)$$
3	2	alrimivv	$$⊢\phi \to \forall x\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)$$
4		ssrel	$$⊢\mathrm{Rel}A\to \left(A\subseteq B\leftrightarrow \forall x\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)\right)$$
5	1 4	syl	$$⊢\phi \to \left(A\subseteq B\leftrightarrow \forall x\forall y\left(⟨x,y⟩\in A\to ⟨x,y⟩\in B\right)\right)$$
6	3 5	mpbird	$$⊢\phi \to A\subseteq B$$