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Theorem setind 4370
 Description: Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
setind (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
Distinct variable group:   𝑥,𝐴

Proof of Theorem setind
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3017 . . . 4 (𝑥𝐴 ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
21imbi1i 237 . . 3 ((𝑥𝐴𝑥𝐴) ↔ (∀𝑦(𝑦𝑥𝑦𝐴) → 𝑥𝐴))
32albii 1405 . 2 (∀𝑥(𝑥𝐴𝑥𝐴) ↔ ∀𝑥(∀𝑦(𝑦𝑥𝑦𝐴) → 𝑥𝐴))
4 setindel 4369 . 2 (∀𝑥(∀𝑦(𝑦𝑥𝑦𝐴) → 𝑥𝐴) → 𝐴 = V)
53, 4sylbi 120 1 (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1288   = wceq 1290   ∈ wcel 1439  Vcvv 2622   ⊆ wss 3002 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4368 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-v 2624  df-in 3008  df-ss 3015 This theorem is referenced by:  setind2  4371
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