Theorem setind2 4356

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 3445	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
2		setind 4355	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
3	1, 2	sylbi 119	1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1287   = wceq 1289   ∈ wcel 1438  Vcvv 2619   ⊆ wss 2999  𝒫 cpw 3429

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431

This theorem is referenced by: (None)