Theorem setind2

Description: Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003)

	Ref	Expression
Assertion	setind2	$$⊢𝒫A\subseteq A\to A=\mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		pwss	$$⊢𝒫A\subseteq A\leftrightarrow \forall x\left(x\subseteq A\to x\in A\right)$$
2		setind	$$⊢\forall x\left(x\subseteq A\to x\in A\right)\to A=\mathrm{V}$$
3	1 2	sylbi	$$⊢𝒫A\subseteq A\to A=\mathrm{V}$$