Theorem ipodrscl

Description: Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015)

	Ref	Expression
Assertion	ipodrscl	$$⊢\mathrm{toInc}\left(A\right)\in \mathrm{Dirset}\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		isipodrs	$$⊢\mathrm{toInc}\left(A\right)\in \mathrm{Dirset}\leftrightarrow \left(A\in \mathrm{V}\wedge A\ne \varnothing \wedge \forall x\in A\forall y\in A\exists z\in Ax\cup y\subseteq z\right)$$
2	1	simp1bi	$$⊢\mathrm{toInc}\left(A\right)\in \mathrm{Dirset}\to A\in \mathrm{V}$$