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Theorem bj-dfom 15163
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 4609 . 2  |-  om  =  |^| { x  |  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
2 df-bj-ind 15157 . . . . 5  |-  (Ind  x  <->  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
)
32bicomi 132 . . . 4  |-  ( (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )  <-> Ind  x )
43abbii 2305 . . 3  |-  { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }  =  {
x  | Ind  x }
54inteqi 3863 . 2  |-  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }  =  |^| { x  | Ind  x }
61, 5eqtri 2210 1  |-  om  =  |^| { x  | Ind  x }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   (/)c0 3437   |^|cint 3859   suc csuc 4383   omcom 4607  Ind wind 15156
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-int 3860  df-iom 4608  df-bj-ind 15157
This theorem is referenced by:  bj-omind  15164  bj-omssind  15165  bj-ssom  15166
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