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Theorem bj-inex 9338
Description: The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inex  V  W  i^i  _V

Proof of Theorem bj-inex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 2562 . 2  V
2 elisset 2562 . 2  W
3 ax-17 1416 . . . 4
4 19.29r 1509 . . . 4
53, 4sylan2 270 . . 3
6 ax-17 1416 . . . . 5
7 19.29 1508 . . . . 5
86, 7sylan 267 . . . 4
98eximi 1488 . . 3
10 ineq12 3127 . . . . 5  i^i  i^i
11102eximi 1489 . . . 4  i^i  i^i
12 dfin5 2919 . . . . . . 7  i^i  {  |  }
13 vex 2554 . . . . . . . 8 
_V
14 ax-bdel 9256 . . . . . . . . 9 BOUNDED
15 bdcv 9283 . . . . . . . . 9 BOUNDED
1614, 15bdrabexg 9337 . . . . . . . 8  _V  {  |  }  _V
1713, 16ax-mp 7 . . . . . . 7  {  |  }  _V
1812, 17eqeltri 2107 . . . . . 6  i^i 
_V
19 eleq1 2097 . . . . . 6  i^i  i^i  i^i  _V  i^i 
_V
2018, 19mpbii 136 . . . . 5  i^i  i^i  i^i 
_V
2120exlimivv 1773 . . . 4  i^i  i^i  i^i  _V
2211, 21syl 14 . . 3  i^i 
_V
235, 9, 223syl 17 . 2  i^i  _V
241, 2, 23syl2an 273 1  V  W  i^i  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242  wex 1378   wcel 1390   {crab 2304   _Vcvv 2551    i^i cin 2910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9248  ax-bdan 9250  ax-bdel 9256  ax-bdsb 9257  ax-bdsep 9319
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-bdc 9276
This theorem is referenced by:  speano5  9378
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