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Theorem bdcint 14565
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 14509 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 14506 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2460 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 14512 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 14537 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3845 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 14532 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  {cab 2163  wral 2455   cint 3844  BOUNDED wbdc 14528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14501  ax-bdal 14506  ax-bdel 14509  ax-bdsb 14510
This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-int 3845  df-bdc 14529
This theorem is referenced by: (None)
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