Theorem bdssexd 15397

Description: Bounded version of ssexd 4169. (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 15396	. 2 |- ( ( A C_ B /\ B e. C ) -> A e. _V )
5	1, 2, 4	syl2anc 411	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 2164   _Vcvv 2760   C_ wss 3153  BOUNDED wbdc 15332

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-bdc 15333

This theorem is referenced by: (None)