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Theorem bj-inex 10841
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  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  B
)  e.  _V )

Proof of Theorem bj-inex
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elisset 2614 . 2  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2614 . 2  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 ax-17 1460 . . . 4  |-  ( E. y  y  =  B  ->  A. x E. y 
y  =  B )
4 19.29r 1553 . . . 4  |-  ( ( E. x  x  =  A  /\  A. x E. y  y  =  B )  ->  E. x
( x  =  A  /\  E. y  y  =  B ) )
53, 4sylan2 280 . . 3  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x ( x  =  A  /\  E. y  y  =  B
) )
6 ax-17 1460 . . . . 5  |-  ( x  =  A  ->  A. y  x  =  A )
7 19.29 1552 . . . . 5  |-  ( ( A. y  x  =  A  /\  E. y 
y  =  B )  ->  E. y ( x  =  A  /\  y  =  B ) )
86, 7sylan 277 . . . 4  |-  ( ( x  =  A  /\  E. y  y  =  B )  ->  E. y
( x  =  A  /\  y  =  B ) )
98eximi 1532 . . 3  |-  ( E. x ( x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
10 ineq12 3163 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  i^i  y
)  =  ( A  i^i  B ) )
11102eximi 1533 . . . 4  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( x  i^i  y
)  =  ( A  i^i  B ) )
12 dfin5 2981 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
13 vex 2605 . . . . . . . 8  |-  x  e. 
_V
14 ax-bdel 10755 . . . . . . . . 9  |- BOUNDED  z  e.  y
15 bdcv 10782 . . . . . . . . 9  |- BOUNDED  x
1614, 15bdrabexg 10840 . . . . . . . 8  |-  ( x  e.  _V  ->  { z  e.  x  |  z  e.  y }  e.  _V )
1713, 16ax-mp 7 . . . . . . 7  |-  { z  e.  x  |  z  e.  y }  e.  _V
1812, 17eqeltri 2152 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
19 eleq1 2142 . . . . . 6  |-  ( ( x  i^i  y )  =  ( A  i^i  B )  ->  ( (
x  i^i  y )  e.  _V  <->  ( A  i^i  B )  e.  _V )
)
2018, 19mpbii 146 . . . . 5  |-  ( ( x  i^i  y )  =  ( A  i^i  B )  ->  ( A  i^i  B )  e.  _V )
2120exlimivv 1818 . . . 4  |-  ( E. x E. y ( x  i^i  y )  =  ( A  i^i  B )  ->  ( A  i^i  B )  e.  _V )
2211, 21syl 14 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  ( A  i^i  B
)  e.  _V )
235, 9, 223syl 17 . 2  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  ( A  i^i  B )  e.  _V )
241, 2, 23syl2an 283 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   {crab 2353   _Vcvv 2602    i^i cin 2973
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 2064  ax-bd0 10747  ax-bdan 10749  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818
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-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-bdc 10775
This theorem is referenced by:  speano5  10882
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