Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcint GIF version

Theorem bdcint 13002
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 12946 . . . . 5 BOUNDED 𝑦𝑧
21ax-bdal 12943 . . . 4 BOUNDED𝑧𝑥 𝑦𝑧
3 df-ral 2398 . . . 4 (∀𝑧𝑥 𝑦𝑧 ↔ ∀𝑧(𝑧𝑥𝑦𝑧))
42, 3bd0 12949 . . 3 BOUNDED𝑧(𝑧𝑥𝑦𝑧)
54bdcab 12974 . 2 BOUNDED {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
6 df-int 3742 . 2 𝑥 = {𝑦 ∣ ∀𝑧(𝑧𝑥𝑦𝑧)}
75, 6bdceqir 12969 1 BOUNDED 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314  {cab 2103  wral 2393   cint 3741  BOUNDED wbdc 12965
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099  ax-bd0 12938  ax-bdal 12943  ax-bdel 12946  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-int 3742  df-bdc 12966
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator