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Theorem setind2 4450
 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 3521 . 2 (𝒫 𝐴𝐴 ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
2 setind 4449 . 2 (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
31, 2sylbi 120 1 (𝒫 𝐴𝐴𝐴 = V)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2681   ⊆ wss 3066  𝒫 cpw 3505 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-setind 4447 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507 This theorem is referenced by: (None)
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