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Theorem bdcdif 12986
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 12971 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . . 6  |- BOUNDED  B
43bdeli 12971 . . . . 5  |- BOUNDED  x  e.  B
54ax-bdn 12942 . . . 4  |- BOUNDED  -.  x  e.  B
62, 5ax-bdan 12940 . . 3  |- BOUNDED  ( x  e.  A  /\  -.  x  e.  B
)
76bdcab 12974 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
8 df-dif 3043 . 2  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
97, 8bdceqir 12969 1  |- BOUNDED  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    e. wcel 1465   {cab 2103    \ cdif 3038  BOUNDED wbdc 12965
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099  ax-bd0 12938  ax-bdan 12940  ax-bdn 12942  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-clab 2104  df-cleq 2110  df-clel 2113  df-dif 3043  df-bdc 12966
This theorem is referenced by:  bdcnulALT  12991
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