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Theorem bdssexg 13550
Description: Bounded version of ssexg 4105. (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 3152 . . . 4  |-  ( x  =  B  ->  ( A  C_  x  <->  A  C_  B
) )
21imbi1d 230 . . 3  |-  ( x  =  B  ->  (
( A  C_  x  ->  A  e.  _V )  <->  ( A  C_  B  ->  A  e.  _V ) ) )
3 bdssexg.bd . . . 4  |- BOUNDED  A
4 vex 2715 . . . 4  |-  x  e. 
_V
53, 4bdssex 13548 . . 3  |-  ( A 
C_  x  ->  A  e.  _V )
62, 5vtoclg 2772 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  e.  _V ) )
76impcom 124 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712    C_ wss 3102  BOUNDED wbdc 13486
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 13530
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 13487
This theorem is referenced by:  bdssexd  13551  bdrabexg  13552  bdunexb  13566
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