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Theorem bdssex 13488
Description: Bounded version of ssex 4101. (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 3115 . 2  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
2 bdssex.bd . . . 4  |- BOUNDED  A
3 bdssex.1 . . . 4  |-  B  e. 
_V
42, 3bdinex2 13486 . . 3  |-  ( A  i^i  B )  e. 
_V
5 eleq1 2220 . . 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 1335    e. wcel 2128   _Vcvv 2712    i^i cin 3101    C_ wss 3102  BOUNDED wbdc 13426
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bdsep 13470
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-bdc 13427
This theorem is referenced by:  bdssexi  13489  bdssexg  13490  bdfind  13532
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