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Theorem bj-ssom 13239
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 3787 . . 3  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A. x  e.  { y  | Ind  y } A  C_  x )
2 df-ral 2421 . . 3  |-  ( A. x  e.  { y  | Ind  y } A  C_  x 
<-> 
A. x ( x  e.  { y  | Ind  y }  ->  A  C_  x ) )
3 vex 2689 . . . . . 6  |-  x  e. 
_V
4 bj-indeq 13232 . . . . . 6  |-  ( y  =  x  ->  (Ind  y 
<-> Ind  x ) )
53, 4elab 2828 . . . . 5  |-  ( x  e.  { y  | Ind  y }  <-> Ind  x )
65imbi1i 237 . . . 4  |-  ( ( x  e.  { y  | Ind  y }  ->  A 
C_  x )  <->  (Ind  x  ->  A  C_  x )
)
76albii 1446 . . 3  |-  ( A. x ( x  e. 
{ y  | Ind  y }  ->  A  C_  x
)  <->  A. x (Ind  x  ->  A  C_  x )
)
81, 2, 73bitrri 206 . 2  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  |^| { y  | Ind  y } )
9 bj-dfom 13236 . . . 4  |-  om  =  |^| { y  | Ind  y }
109eqcomi 2143 . . 3  |-  |^| { y  | Ind  y }  =  om
1110sseq2i 3124 . 2  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A  C_  om )
128, 11bitri 183 1  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329    e. wcel 1480   {cab 2125   A.wral 2416    C_ wss 3071   |^|cint 3771   omcom 4504  Ind wind 13229
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-int 3772  df-iom 4505  df-bj-ind 13230
This theorem is referenced by:  bj-om  13240
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