Theorem bdssexd 13274

Description: Bounded version of ssexd 4076. (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 13273	. 2 |- ( ( A C_ B /\ B e. C ) -> A e. _V )
5	1, 2, 4	syl2anc 409	1 |- ( ph -> A e. _V )

Syntax hints:   -> wi 4   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: (None)