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Theorem bdcin 12895
 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 12878 . . . 4 BOUNDED 𝑥𝐴
3 bdcdif.2 . . . . 5 BOUNDED 𝐵
43bdeli 12878 . . . 4 BOUNDED 𝑥𝐵
52, 4ax-bdan 12847 . . 3 BOUNDED (𝑥𝐴𝑥𝐵)
65bdcab 12881 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
7 df-in 3045 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
86, 7bdceqir 12876 1 BOUNDED (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∈ wcel 1463  {cab 2101   ∩ cin 3038  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-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12845  ax-bdan 12847  ax-bdsb 12854 This theorem depends on definitions:  df-bi 116  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-bdc 12873 This theorem is referenced by: (None)
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