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Theorem bdssex 13784
Description: Bounded version of ssex 4119. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdssex.bd BOUNDED 𝐴
bdssex.1 𝐵 ∈ V
Assertion
Ref Expression
bdssex (𝐴𝐵𝐴 ∈ V)

Proof of Theorem bdssex
StepHypRef Expression
1 df-ss 3129 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 bdssex.bd . . . 4 BOUNDED 𝐴
3 bdssex.1 . . . 4 𝐵 ∈ V
42, 3bdinex2 13782 . . 3 (𝐴𝐵) ∈ V
5 eleq1 2229 . . 3 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ V ↔ 𝐴 ∈ V))
64, 5mpbii 147 . 2 ((𝐴𝐵) = 𝐴𝐴 ∈ V)
71, 6sylbi 120 1 (𝐴𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1343  wcel 2136  Vcvv 2726  cin 3115  wss 3116  BOUNDED wbdc 13722
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13766
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13723
This theorem is referenced by:  bdssexi  13785  bdssexg  13786  bdfind  13828
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