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Theorem bj-indint 10884
Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indint  |- Ind  |^| { x  e.  A  | Ind  x }
Distinct variable group:    x, A

Proof of Theorem bj-indint
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 10880 . . . . 5  |-  (Ind  x  <->  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
)
21simplbi 268 . . . 4  |-  (Ind  x  -> 
(/)  e.  x )
32rgenw 2419 . . 3  |-  A. x  e.  A  (Ind  x  -> 
(/)  e.  x )
4 0ex 3913 . . . 4  |-  (/)  e.  _V
54elintrab 3656 . . 3  |-  ( (/)  e.  |^| { x  e.  A  | Ind  x }  <->  A. x  e.  A  (Ind  x  ->  (/)  e.  x
) )
63, 5mpbir 144 . 2  |-  (/)  e.  |^| { x  e.  A  | Ind  x }
7 bj-indsuc 10881 . . . . . 6  |-  (Ind  x  ->  ( y  e.  x  ->  suc  y  e.  x
) )
87a2i 11 . . . . 5  |-  ( (Ind  x  ->  y  e.  x )  ->  (Ind  x  ->  suc  y  e.  x ) )
98ralimi 2427 . . . 4  |-  ( A. x  e.  A  (Ind  x  ->  y  e.  x
)  ->  A. x  e.  A  (Ind  x  ->  suc  y  e.  x
) )
10 vex 2605 . . . . 5  |-  y  e. 
_V
1110elintrab 3656 . . . 4  |-  ( y  e.  |^| { x  e.  A  | Ind  x }  <->  A. x  e.  A  (Ind  x  ->  y  e.  x ) )
1210bj-sucex 10872 . . . . 5  |-  suc  y  e.  _V
1312elintrab 3656 . . . 4  |-  ( suc  y  e.  |^| { x  e.  A  | Ind  x } 
<-> 
A. x  e.  A  (Ind  x  ->  suc  y  e.  x ) )
149, 11, 133imtr4i 199 . . 3  |-  ( y  e.  |^| { x  e.  A  | Ind  x }  ->  suc  y  e.  |^| { x  e.  A  | Ind  x } )
1514rgen 2417 . 2  |-  A. y  e.  |^| { x  e.  A  | Ind  x } suc  y  e.  |^| { x  e.  A  | Ind  x }
16 df-bj-ind 10880 . 2  |-  (Ind  |^| { x  e.  A  | Ind  x }  <->  ( (/)  e.  |^| { x  e.  A  | Ind  x }  /\  A. y  e.  |^| { x  e.  A  | Ind  x } suc  y  e.  |^| { x  e.  A  | Ind  x } ) )
176, 15, 16mpbir2an 884 1  |- Ind  |^| { x  e.  A  | Ind  x }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   A.wral 2349   {crab 2353   (/)c0 3258   |^|cint 3644   suc csuc 4128  Ind wind 10879
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-in1 577  ax-in2 578  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-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912  ax-pr 3972  ax-un 4196  ax-bd0 10762  ax-bdor 10765  ax-bdex 10768  ax-bdeq 10769  ax-bdel 10770  ax-bdsb 10771  ax-bdsep 10833
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-bdc 10790  df-bj-ind 10880
This theorem is referenced by:  bj-omind  10887
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