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Theorem setind2 4265
 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 𝐴𝐴𝐴 = V)

Proof of Theorem setind2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwss 3374 . 2 (𝒫 𝐴𝐴 ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
2 setind 4264 . 2 (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
31, 2sylbi 114 1 (𝒫 𝐴𝐴𝐴 = V)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361 This theorem is referenced by: (None)
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