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	⊢ ( 𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V )

Proof

Step	Hyp	Ref	Expression
1		pwss	⊢ ( 𝒫 𝐴 ⊆ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴 ) )
2		setind	⊢ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴 ) → 𝐴 = V )
3	1 2	sylbi	⊢ ( 𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V )