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Theorem bdssexg 14592
Description: Bounded version of ssexg 4142. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdssexg.bd  |- BOUNDED  A
Assertion
Ref Expression
bdssexg  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )

Proof of Theorem bdssexg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3179 . . . 4  |-  ( x  =  B  ->  ( A  C_  x  <->  A  C_  B
) )
21imbi1d 231 . . 3  |-  ( x  =  B  ->  (
( A  C_  x  ->  A  e.  _V )  <->  ( A  C_  B  ->  A  e.  _V ) ) )
3 bdssexg.bd . . . 4  |- BOUNDED  A
4 vex 2740 . . . 4  |-  x  e. 
_V
53, 4bdssex 14590 . . 3  |-  ( A 
C_  x  ->  A  e.  _V )
62, 5vtoclg 2797 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  e.  _V ) )
76impcom 125 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129  BOUNDED wbdc 14528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsep 14572
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-bdc 14529
This theorem is referenced by:  bdssexd  14593  bdrabexg  14594  bdunexb  14608
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