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Theorem bdcint 13064
Description: The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
Assertion
Ref Expression
bdcint BOUNDED 𝑥

Proof of Theorem bdcint
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdel 13008 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 13005 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2419 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 13011 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 13036 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3767 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 13031 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  {cab 2123  wral 2414   cint 3766  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-bdal 13005  ax-bdel 13008  ax-bdsb 13009
This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-int 3767  df-bdc 13028
This theorem is referenced by: (None)
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