Theorem bdssexg 16691

Description: Bounded version of ssexg 4251. (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 3264	. . . 4 |- ( x = B -> ( A C_ x <-> A C_ B ) )
2	1	imbi1d 231	. . 3 |- ( x = B -> ( ( A C_ x -> A e. _V ) <-> ( A C_ B -> A e. _V ) ) )
3		bdssexg.bd	. . . 4 |- BOUNDED A
4		vex 2818	. . . 4 |- x e. _V
5	3, 4	bdssex 16689	. . 3 |- ( A C_ x -> A e. _V )
6	2, 5	vtoclg 2877	. 2 |- ( B e. C -> ( A C_ B -> A e. _V ) )
7	6	impcom 125	1 |- ( ( A C_ B /\ B e. C ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213  BOUNDED wbdc 16627

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-bdsep 16671

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-ss 3226  df-bdc 16628

This theorem is referenced by:  bdssexd  16692  bdrabexg  16693  bdunexb  16707