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Theorem bdcdif 9981
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 9966 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . . 6  |- BOUNDED  B
43bdeli 9966 . . . . 5  |- BOUNDED  x  e.  B
54ax-bdn 9937 . . . 4  |- BOUNDED  -.  x  e.  B
62, 5ax-bdan 9935 . . 3  |- BOUNDED  ( x  e.  A  /\  -.  x  e.  B
)
76bdcab 9969 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
8 df-dif 2920 . 2  |-  ( A 
\  B )  =  { x  |  ( x  e.  A  /\  -.  x  e.  B
) }
97, 8bdceqir 9964 1  |- BOUNDED  ( A  \  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 97    e. wcel 1393   {cab 2026    \ cdif 2914  BOUNDED wbdc 9960
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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdn 9937  ax-bdsb 9942
This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-dif 2920  df-bdc 9961
This theorem is referenced by:  bdcnulALT  9986
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