Theorem bdssex 13784

Description: Bounded version of ssex 4119. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssex.bd	|- BOUNDED A
bdssex.1	|- B e. _V

Assertion

Ref	Expression
bdssex	|- ( A C_ B -> A e. _V )

Proof of Theorem bdssex

Step	Hyp	Ref	Expression
1		df-ss 3129	. 2 |- ( A C_ B <-> ( A i^i B ) = A )
2		bdssex.bd	. . . 4 |- BOUNDED A
3		bdssex.1	. . . 4 |- B e. _V
4	2, 3	bdinex2 13782	. . 3 |- ( A i^i B ) e. _V
5		eleq1 2229	. . 3 |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) )
6	4, 5	mpbii 147	. 2 |- ( ( A i^i B ) = A -> A e. _V )
7	1, 6	sylbi 120	1 |- ( A C_ B -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116  BOUNDED wbdc 13722

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13723

This theorem is referenced by:  bdssexi  13785  bdssexg  13786  bdfind  13828