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Theorem bdcdif 15298
Description: The difference 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
bdcdif  |- BOUNDED  ( A  \  B
)

Proof of Theorem bdcdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 bdcdif.1 . . . . 5  |- BOUNDED  A
21bdeli 15283 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . . 6  |- BOUNDED  B
43bdeli 15283 . . . . 5  |- BOUNDED  x  e.  B
54ax-bdn 15254 . . . 4  |- BOUNDED  -.  x  e.  B
62, 5ax-bdan 15252 . . 3  |- BOUNDED  ( x  e.  A  /\  -.  x  e.  B
)
76bdcab 15286 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
8 df-dif 3155 . 2  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
97, 8bdceqir 15281 1  |- BOUNDED  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    e. wcel 2164   {cab 2179    \ cdif 3150  BOUNDED wbdc 15277
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 2175  ax-bd0 15250  ax-bdan 15252  ax-bdn 15254  ax-bdsb 15259
This theorem depends on definitions:  df-bi 117  df-clab 2180  df-cleq 2186  df-clel 2189  df-dif 3155  df-bdc 15278
This theorem is referenced by:  bdcnulALT  15303
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