Theorem bdssexg 13796

Description: Bounded version of ssexg 4121. (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.

Step	Hyp	Ref	Expression
1		sseq2 3166	. . . 4 |- ( x = B -> ( A C_ x <-> A C_ B ) )
2	1	imbi1d 230	. . 3 |- ( x = B -> ( ( A C_ x -> A e. _V ) <-> ( A C_ B -> A e. _V ) ) )
3		bdssexg.bd	. . . 4 |- BOUNDED A
4		vex 2729	. . . 4 |- x e. _V
5	3, 4	bdssex 13794	. . 3 |- ( A C_ x -> A e. _V )
6	2, 5	vtoclg 2786	. 2 |- ( B e. C -> ( A C_ B -> A e. _V ) )
7	6	impcom 124	1 |- ( ( A C_ B /\ B e. C ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116  BOUNDED wbdc 13732

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13776

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13733

This theorem is referenced by:  bdssexd  13797  bdrabexg  13798  bdunexb  13812