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Theorem bdssex 13937
Description: Bounded version of ssex 4126. (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 3134 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 bdssex.bd . . . 4 BOUNDED 𝐴
3 bdssex.1 . . . 4 𝐵 ∈ V
42, 3bdinex2 13935 . . 3 (𝐴𝐵) ∈ V
5 eleq1 2233 . . 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 1348  wcel 2141  Vcvv 2730  cin 3120  wss 3121  BOUNDED wbdc 13875
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-bdc 13876
This theorem is referenced by:  bdssexi  13938  bdssexg  13939  bdfind  13981
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