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Theorem bdcdif 13048
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 13033 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . . 6  |- BOUNDED  B
43bdeli 13033 . . . . 5  |- BOUNDED  x  e.  B
54ax-bdn 13004 . . . 4  |- BOUNDED  -.  x  e.  B
62, 5ax-bdan 13002 . . 3  |- BOUNDED  ( x  e.  A  /\  -.  x  e.  B
)
76bdcab 13036 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
8 df-dif 3068 . 2  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
97, 8bdceqir 13031 1  |- BOUNDED  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    e. wcel 1480   {cab 2123    \ cdif 3063  BOUNDED wbdc 13027
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdan 13002  ax-bdn 13004  ax-bdsb 13009
This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-dif 3068  df-bdc 13028
This theorem is referenced by:  bdcnulALT  13053
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