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Theorem bdcdif 13089
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 (𝐴𝐵)

Proof of Theorem bdcdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bdcdif.1 . . . . 5 BOUNDED 𝐴
21bdeli 13074 . . . 4 BOUNDED 𝑥𝐴
3 bdcdif.2 . . . . . 6 BOUNDED 𝐵
43bdeli 13074 . . . . 5 BOUNDED 𝑥𝐵
54ax-bdn 13045 . . . 4 BOUNDED ¬ 𝑥𝐵
62, 5ax-bdan 13043 . . 3 BOUNDED (𝑥𝐴 ∧ ¬ 𝑥𝐵)
76bdcab 13077 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
8 df-dif 3073 . 2 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
97, 8bdceqir 13072 1 BOUNDED (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wcel 1480  {cab 2125  cdif 3068  BOUNDED wbdc 13068
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 2121  ax-bd0 13041  ax-bdan 13043  ax-bdn 13045  ax-bdsb 13050
This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-dif 3073  df-bdc 13069
This theorem is referenced by:  bdcnulALT  13094
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