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Theorem bdssex 13937
Description: Bounded version of ssex 4126. (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 3134 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 bdssex.bd . . . 4  |- BOUNDED  A
3 bdssex.1 . . . 4  |-  B  e. 
_V
42, 3bdinex2 13935 . . 3  |-  ( A  i^i  B )  e. 
_V
5 eleq1 2233 . . 3  |-  ( ( A  i^i  B )  =  A  ->  (
( A  i^i  B
)  e.  _V  <->  A  e.  _V ) )
64, 5mpbii 147 . 2  |-  ( ( A  i^i  B )  =  A  ->  A  e.  _V )
71, 6sylbi 120 1  |-  ( A 
C_  B  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   _Vcvv 2730    i^i cin 3120    C_ wss 3121  BOUNDED wbdc 13875
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-bdc 13876
This theorem is referenced by:  bdssexi  13938  bdssexg  13939  bdfind  13981
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