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Theorem bj-indint 12940
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 12936 . . . . 5  |-  (Ind  x  <->  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
)
21simplbi 270 . . . 4  |-  (Ind  x  -> 
(/)  e.  x )
32rgenw 2462 . . 3  |-  A. x  e.  A  (Ind  x  -> 
(/)  e.  x )
4 0ex 4023 . . . 4  |-  (/)  e.  _V
54elintrab 3751 . . 3  |-  ( (/)  e.  |^| { x  e.  A  | Ind  x }  <->  A. x  e.  A  (Ind  x  ->  (/)  e.  x
) )
63, 5mpbir 145 . 2  |-  (/)  e.  |^| { x  e.  A  | Ind  x }
7 bj-indsuc 12937 . . . . . 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 2470 . . . 4  |-  ( A. x  e.  A  (Ind  x  ->  y  e.  x
)  ->  A. x  e.  A  (Ind  x  ->  suc  y  e.  x
) )
10 vex 2661 . . . . 5  |-  y  e. 
_V
1110elintrab 3751 . . . 4  |-  ( y  e.  |^| { x  e.  A  | Ind  x }  <->  A. x  e.  A  (Ind  x  ->  y  e.  x ) )
1210bj-sucex 12932 . . . . 5  |-  suc  y  e.  _V
1312elintrab 3751 . . . 4  |-  ( suc  y  e.  |^| { x  e.  A  | Ind  x } 
<-> 
A. x  e.  A  (Ind  x  ->  suc  y  e.  x ) )
149, 11, 133imtr4i 200 . . 3  |-  ( y  e.  |^| { x  e.  A  | Ind  x }  ->  suc  y  e.  |^| { x  e.  A  | Ind  x } )
1514rgen 2460 . 2  |-  A. y  e.  |^| { x  e.  A  | Ind  x } suc  y  e.  |^| { x  e.  A  | Ind  x }
16 df-bj-ind 12936 . 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 909 1  |- Ind  |^| { x  e.  A  | Ind  x }
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   A.wral 2391   {crab 2395   (/)c0 3331   |^|cint 3739   suc csuc 4255  Ind wind 12935
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12822  ax-bdor 12825  ax-bdex 12828  ax-bdeq 12829  ax-bdel 12830  ax-bdsb 12831  ax-bdsep 12893
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-bdc 12850  df-bj-ind 12936
This theorem is referenced by:  bj-omind  12943
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