Theorem setind2 4524

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 3582	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
2		setind 4523	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
3	1, 2	sylbi 120	1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1346   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by: (None)