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Theorem bdcint 10826
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 10770 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 10767 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2354 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 10773 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 10798 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3645 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 10793 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  {cab 2068  wral 2349   cint 3644  BOUNDED wbdc 10789
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064  ax-bd0 10762  ax-bdal 10767  ax-bdel 10770  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-int 3645  df-bdc 10790
This theorem is referenced by: (None)
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