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Theorem bj-ssom 10998
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 3672 . . 3  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A. x  e.  { y  | Ind  y } A  C_  x )
2 df-ral 2358 . . 3  |-  ( A. x  e.  { y  | Ind  y } A  C_  x 
<-> 
A. x ( x  e.  { y  | Ind  y }  ->  A  C_  x ) )
3 vex 2613 . . . . . 6  |-  x  e. 
_V
4 bj-indeq 10991 . . . . . 6  |-  ( y  =  x  ->  (Ind  y 
<-> Ind  x ) )
53, 4elab 2746 . . . . 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 1400 . . 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 10995 . . . 4  |-  om  =  |^| { y  | Ind  y }
109eqcomi 2087 . . 3  |-  |^| { y  | Ind  y }  =  om
1110sseq2i 3033 . 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 1283    e. wcel 1434   {cab 2069   A.wral 2353    C_ wss 2982   |^|cint 3656   omcom 4359  Ind wind 10988
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-in 2988  df-ss 2995  df-int 3657  df-iom 4360  df-bj-ind 10989
This theorem is referenced by:  bj-om  10999
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