Theorem bdssexd 12069

Description: Bounded version of ssexd 3985. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssexd.1	|- ( ph -> B e. C )
bdssexd.2	|- ( ph -> A C_ B )
bdssexd.bd	|- BOUNDED A

Assertion

Ref	Expression
bdssexd	|- ( ph -> A e. _V )

Proof of Theorem bdssexd

Step	Hyp	Ref	Expression
1		bdssexd.2	. 2 |- ( ph -> A C_ B )
2		bdssexd.1	. 2 |- ( ph -> B e. C )
3		bdssexd.bd	. . 3 |- BOUNDED A
4	3	bdssexg 12068	. 2 |- ( ( A C_ B /\ B e. C ) -> A e. _V )
5	1, 2, 4	syl2anc 404	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 1439   _Vcvv 2620   C_ wss 3000  BOUNDED wbdc 12004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-bdsep 12048

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006  df-ss 3013  df-bdc 12005

This theorem is referenced by: (None)