Theorem bdssexg 13102

Description: Bounded version of ssexg 4067. (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 3121	. . . 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 2689	. . . 4 |- x e. _V
5	3, 4	bdssex 13100	. . 3 |- ( A C_ x -> A e. _V )
6	2, 5	vtoclg 2746	. 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 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071  BOUNDED wbdc 13038

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-bdc 13039

This theorem is referenced by:  bdssexd  13103  bdrabexg  13104  bdunexb  13118