Theorem setind2 4463

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 3531	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴))
2		setind 4462	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
3	1, 2	sylbi 120	1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1330   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ⊆ wss 3076  𝒫 cpw 3515

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by: (None)