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Theorem bdssex 10851
Description: Bounded version of ssex 3923. (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 2987 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 bdssex.bd . . . 4  |- BOUNDED  A
3 bdssex.1 . . . 4  |-  B  e. 
_V
42, 3bdinex2 10849 . . 3  |-  ( A  i^i  B )  e. 
_V
5 eleq1 2142 . . 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 1285    e. wcel 1434   _Vcvv 2602    i^i cin 2973    C_ wss 2974  BOUNDED wbdc 10789
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-bdsep 10833
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-v 2604  df-in 2980  df-ss 2987  df-bdc 10790
This theorem is referenced by:  bdssexi  10852  bdssexg  10853  bdfind  10899
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