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Theorem bdssexg 10024
Description: Bounded version of ssexg 3896. (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 2967 . . . 4  |-  ( x  =  B  ->  ( A  C_  x  <->  A  C_  B
) )
21imbi1d 220 . . 3  |-  ( x  =  B  ->  (
( A  C_  x  ->  A  e.  _V )  <->  ( A  C_  B  ->  A  e.  _V ) ) )
3 bdssexg.bd . . . 4  |- BOUNDED  A
4 vex 2560 . . . 4  |-  x  e. 
_V
53, 4bdssex 10022 . . 3  |-  ( A 
C_  x  ->  A  e.  _V )
62, 5vtoclg 2613 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  e.  _V ) )
76impcom 116 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557    C_ wss 2917  BOUNDED wbdc 9960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961
This theorem is referenced by:  bdssexd  10025  bdrabexg  10026  bdunexb  10040
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