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Theorem bdcint 11425
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 11369 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 11366 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2364 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 11372 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 11397 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3684 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 11392 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  {cab 2074  wral 2359   cint 3683  BOUNDED wbdc 11388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11361  ax-bdal 11366  ax-bdel 11369  ax-bdsb 11370
This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-int 3684  df-bdc 11389
This theorem is referenced by: (None)
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