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Theorem bj-inex 12939
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 2672 . 2  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2672 . 2  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 ax-17 1489 . . . 4  |-  ( E. y  y  =  B  ->  A. x E. y 
y  =  B )
4 19.29r 1583 . . . 4  |-  ( ( E. x  x  =  A  /\  A. x E. y  y  =  B )  ->  E. x
( x  =  A  /\  E. y  y  =  B ) )
53, 4sylan2 282 . . 3  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x ( x  =  A  /\  E. y  y  =  B
) )
6 ax-17 1489 . . . . 5  |-  ( x  =  A  ->  A. y  x  =  A )
7 19.29 1582 . . . . 5  |-  ( ( A. y  x  =  A  /\  E. y 
y  =  B )  ->  E. y ( x  =  A  /\  y  =  B ) )
86, 7sylan 279 . . . 4  |-  ( ( x  =  A  /\  E. y  y  =  B )  ->  E. y
( x  =  A  /\  y  =  B ) )
98eximi 1562 . . 3  |-  ( E. x ( x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
10 ineq12 3240 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  i^i  y
)  =  ( A  i^i  B ) )
11102eximi 1563 . . . 4  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( x  i^i  y
)  =  ( A  i^i  B ) )
12 dfin5 3046 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
13 vex 2661 . . . . . . . 8  |-  x  e. 
_V
14 ax-bdel 12853 . . . . . . . . 9  |- BOUNDED  z  e.  y
15 bdcv 12880 . . . . . . . . 9  |- BOUNDED  x
1614, 15bdrabexg 12938 . . . . . . . 8  |-  ( x  e.  _V  ->  { z  e.  x  |  z  e.  y }  e.  _V )
1713, 16ax-mp 5 . . . . . . 7  |-  { z  e.  x  |  z  e.  y }  e.  _V
1812, 17eqeltri 2188 . . . . . 6  |-  ( x  i^i  y )  e. 
_V
19 eleq1 2178 . . . . . 6  |-  ( ( x  i^i  y )  =  ( A  i^i  B )  ->  ( (
x  i^i  y )  e.  _V  <->  ( A  i^i  B )  e.  _V )
)
2018, 19mpbii 147 . . . . 5  |-  ( ( x  i^i  y )  =  ( A  i^i  B )  ->  ( A  i^i  B )  e.  _V )
2120exlimivv 1850 . . . 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 285 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  i^i  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   {crab 2395   _Vcvv 2658    i^i cin 3038
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-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-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12845  ax-bdan 12847  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916
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-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-bdc 12873
This theorem is referenced by:  speano5  12976
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