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Theorem bdcin 11754
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 11737 . . . 4 BOUNDED 𝑥𝐴
3 bdcdif.2 . . . . 5 BOUNDED 𝐵
43bdeli 11737 . . . 4 BOUNDED 𝑥𝐵
52, 4ax-bdan 11706 . . 3 BOUNDED (𝑥𝐴𝑥𝐵)
65bdcab 11740 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
7 df-in 3005 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
86, 7bdceqir 11735 1 BOUNDED (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wcel 1438  {cab 2074  cin 2998  BOUNDED wbdc 11731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11704  ax-bdan 11706  ax-bdsb 11713
This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-bdc 11732
This theorem is referenced by: (None)
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