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Theorem bdssex 11439
Description: Bounded version of ssex 3968. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdssex.bd  |- BOUNDED  A
bdssex.1  |-  B  e. 
_V
Assertion
Ref Expression
bdssex  |-  ( A 
C_  B  ->  A  e.  _V )

Proof of Theorem bdssex
StepHypRef Expression
1 df-ss 3010 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 bdssex.bd . . . 4  |- BOUNDED  A
3 bdssex.1 . . . 4  |-  B  e. 
_V
42, 3bdinex2 11437 . . 3  |-  ( A  i^i  B )  e. 
_V
5 eleq1 2150 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
64, 5mpbii 146 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
71, 6sylbi 119 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619    i^i cin 2996    C_ wss 2997  BOUNDED wbdc 11377
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bdsep 11421
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-bdc 11378
This theorem is referenced by:  bdssexi  11440  bdssexg  11441  bdfind  11487
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