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Theorem bdcint 9997
 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 9941 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 9938 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2311 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 9944 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 9969 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3616 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 9964 1 BOUNDED 𝑥
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  {cab 2026  ∀wral 2306  ∩ cint 3615  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-bdal 9938  ax-bdel 9941  ax-bdsb 9942 This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-int 3616  df-bdc 9961 This theorem is referenced by: (None)
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