Theorem bdssexg 13273

Description: Bounded version of ssexg 4075. (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 3126	. . . 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 2692	. . . 4 |- x e. _V
5	3, 4	bdssex 13271	. . 3 |- ( A C_ x -> A e. _V )
6	2, 5	vtoclg 2749	. 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 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076  BOUNDED wbdc 13209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bdsep 13253

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-bdc 13210

This theorem is referenced by:  bdssexd  13274  bdrabexg  13275  bdunexb  13289