Theorem setind2 4609

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 3645	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
2		setind 4608	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
3	1, 2	sylbi 121	1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1373   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ⊆ wss 3177  𝒫 cpw 3629

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-ral 2493  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631

This theorem is referenced by: (None)