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Theorem bdcin 9298
 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 9281 . . . 4 BOUNDED x A
3 bdcdif.2 . . . . 5 BOUNDED B
43bdeli 9281 . . . 4 BOUNDED x B
52, 4ax-bdan 9250 . . 3 BOUNDED (x A x B)
65bdcab 9284 . 2 BOUNDED {x ∣ (x A x B)}
7 df-in 2918 . 2 (AB) = {x ∣ (x A x B)}
86, 7bdceqir 9279 1 BOUNDED (AB)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∈ wcel 1390  {cab 2023   ∩ cin 2910  BOUNDED wbdc 9275 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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019  ax-bd0 9248  ax-bdan 9250  ax-bdsb 9257 This theorem depends on definitions:  df-bi 110  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-bdc 9276 This theorem is referenced by: (None)
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