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Theorem bj-ssom 11477
Description: A characterization of subclasses of  om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ssom  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
Distinct variable group:    x, A

Proof of Theorem bj-ssom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssint 3699 . . 3  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A. x  e.  { y  | Ind  y } A  C_  x )
2 df-ral 2364 . . 3  |-  ( A. x  e.  { y  | Ind  y } A  C_  x 
<-> 
A. x ( x  e.  { y  | Ind  y }  ->  A  C_  x ) )
3 vex 2622 . . . . . 6  |-  x  e. 
_V
4 bj-indeq 11470 . . . . . 6  |-  ( y  =  x  ->  (Ind  y 
<-> Ind  x ) )
53, 4elab 2758 . . . . 5  |-  ( x  e.  { y  | Ind  y }  <-> Ind  x )
65imbi1i 236 . . . 4  |-  ( ( x  e.  { y  | Ind  y }  ->  A 
C_  x )  <->  (Ind  x  ->  A  C_  x )
)
76albii 1404 . . 3  |-  ( A. x ( x  e. 
{ y  | Ind  y }  ->  A  C_  x
)  <->  A. x (Ind  x  ->  A  C_  x )
)
81, 2, 73bitrri 205 . 2  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  |^| { y  | Ind  y } )
9 bj-dfom 11474 . . . 4  |-  om  =  |^| { y  | Ind  y }
109eqcomi 2092 . . 3  |-  |^| { y  | Ind  y }  =  om
1110sseq2i 3049 . 2  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A  C_  om )
128, 11bitri 182 1  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    e. wcel 1438   {cab 2074   A.wral 2359    C_ wss 2997   |^|cint 3683   omcom 4395  Ind wind 11467
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3003  df-ss 3010  df-int 3684  df-iom 4396  df-bj-ind 11468
This theorem is referenced by:  bj-om  11478
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