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Theorem bdssex 12934
Description: Bounded version of ssex 4033. (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 3052 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 bdssex.bd . . . 4 BOUNDED 𝐴
3 bdssex.1 . . . 4 𝐵 ∈ V
42, 3bdinex2 12932 . . 3 (𝐴𝐵) ∈ V
5 eleq1 2178 . . 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 1314  wcel 1463  Vcvv 2658  cin 3038  wss 3039  BOUNDED wbdc 12872
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-bdc 12873
This theorem is referenced by:  bdssexi  12935  bdssexg  12936  bdfind  12978
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