Theorem bdssexg 16620

Description: Bounded version of ssexg 4233. (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 3252	. . . 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 2806	. . . 4 |- x e. _V
5	3, 4	bdssex 16618	. . 3 |- ( A C_ x -> A e. _V )
6	2, 5	vtoclg 2865	. 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 2202   _Vcvv 2803   C_ wss 3201  BOUNDED wbdc 16556

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 2213  ax-bdsep 16600

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-bdc 16557

This theorem is referenced by:  bdssexd  16621  bdrabexg  16622  bdunexb  16636