Theorem setind2 4661

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

Assertion

Ref	Expression
setind2	⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Proof of Theorem setind2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwss 3687	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
2		setind 4660	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
3	1, 2	sylbi 121	1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1396   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  𝒫 cpw 3668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670

This theorem is referenced by: (None)