Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdssexg Unicode version

Theorem bdssexg 12925
Description: Bounded version of ssexg 4035. (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 3089 . . . 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 2661 . . . 4  |-  x  e. 
_V
53, 4bdssex 12923 . . 3  |-  ( A 
C_  x  ->  A  e.  _V )
62, 5vtoclg 2718 . 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 1314    e. wcel 1463   _Vcvv 2658    C_ wss 3039  BOUNDED wbdc 12861
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdsep 12905
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-bdc 12862
This theorem is referenced by:  bdssexd  12926  bdrabexg  12927  bdunexb  12941
  Copyright terms: Public domain W3C validator