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Theorem bj-dfom 11258
Description: Alternate definition of  om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-dfom  |-  om  =  |^| { x  | Ind  x }

Proof of Theorem bj-dfom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfom3 4379 . 2  |-  om  =  |^| { x  |  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
2 df-bj-ind 11252 . . . . 5  |-  (Ind  x  <->  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
)
32bicomi 130 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  <-> Ind  x )
43abbii 2200 . . 3  |-  { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }  =  {
x  | Ind  x }
54inteqi 3675 . 2  |-  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }  =  |^| { x  | Ind  x }
61, 5eqtri 2105 1  |-  om  =  |^| { x  | Ind  x }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   (/)c0 3275   |^|cint 3671   suc csuc 4165   omcom 4377  Ind wind 11251
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-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-int 3672  df-iom 4378  df-bj-ind 11252
This theorem is referenced by:  bj-omind  11259  bj-omssind  11260  bj-ssom  11261
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