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Theorem bdcin 15068
Description: The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypotheses
Ref Expression
bdcdif.1 BOUNDED 𝐴
bdcdif.2 BOUNDED 𝐵
Assertion
Ref Expression
bdcin BOUNDED (𝐴𝐵)

Proof of Theorem bdcin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bdcdif.1 . . . . 5 BOUNDED 𝐴
21bdeli 15051 . . . 4 BOUNDED 𝑥𝐴
3 bdcdif.2 . . . . 5 BOUNDED 𝐵
43bdeli 15051 . . . 4 BOUNDED 𝑥𝐵
52, 4ax-bdan 15020 . . 3 BOUNDED (𝑥𝐴𝑥𝐵)
65bdcab 15054 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
7 df-in 3150 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
86, 7bdceqir 15049 1 BOUNDED (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2160  {cab 2175  cin 3143  BOUNDED wbdc 15045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15018  ax-bdan 15020  ax-bdsb 15027
This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-bdc 15046
This theorem is referenced by: (None)
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