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Theorem bdssex 10022
 Description: Bounded version of ssex 3894. (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 2931 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 bdssex.bd . . . 4 BOUNDED 𝐴
3 bdssex.1 . . . 4 𝐵 ∈ V
42, 3bdinex2 10020 . . 3 (𝐴𝐵) ∈ V
5 eleq1 2100 . . 3 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ V ↔ 𝐴 ∈ V))
64, 5mpbii 136 . 2 ((𝐴𝐵) = 𝐴𝐴 ∈ V)
71, 6sylbi 114 1 (𝐴𝐵𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917  BOUNDED wbdc 9960 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-bdsep 10004 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-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961 This theorem is referenced by:  bdssexi  10023  bdssexg  10024  bdfind  10071
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