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Theorem bj-ssom 15166
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 3875 . . 3  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A. x  e.  { y  | Ind  y } A  C_  x )
2 df-ral 2473 . . 3  |-  ( A. x  e.  { y  | Ind  y } A  C_  x 
<-> 
A. x ( x  e.  { y  | Ind  y }  ->  A  C_  x ) )
3 vex 2755 . . . . . 6  |-  x  e. 
_V
4 bj-indeq 15159 . . . . . 6  |-  ( y  =  x  ->  (Ind  y 
<-> Ind  x ) )
53, 4elab 2896 . . . . 5  |-  ( x  e.  { y  | Ind  y }  <-> Ind  x )
65imbi1i 238 . . . 4  |-  ( ( x  e.  { y  | Ind  y }  ->  A 
C_  x )  <->  (Ind  x  ->  A  C_  x )
)
76albii 1481 . . 3  |-  ( A. x ( x  e. 
{ y  | Ind  y }  ->  A  C_  x
)  <->  A. x (Ind  x  ->  A  C_  x )
)
81, 2, 73bitrri 207 . 2  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  |^| { y  | Ind  y } )
9 bj-dfom 15163 . . . 4  |-  om  =  |^| { y  | Ind  y }
109eqcomi 2193 . . 3  |-  |^| { y  | Ind  y }  =  om
1110sseq2i 3197 . 2  |-  ( A 
C_  |^| { y  | Ind  y }  <->  A  C_  om )
128, 11bitri 184 1  |-  ( A. x (Ind  x  ->  A 
C_  x )  <->  A  C_  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362    e. wcel 2160   {cab 2175   A.wral 2468    C_ wss 3144   |^|cint 3859   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-v 2754  df-in 3150  df-ss 3157  df-int 3860  df-iom 4608  df-bj-ind 15157
This theorem is referenced by:  bj-om  15167
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