Theorem bdssexd 10025

Description: Bounded version of ssexd 3897. (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 10024	. 2 |- ( ( A C_ B /\ B e. C ) -> A e. _V )
5	1, 2, 4	syl2anc 391	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 1393   _Vcvv 2557   C_ wss 2917  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by: (None)