Theorem bdssex 16672

Description: Bounded version of ssex 4247. (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 3224	. 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 16670	. . 3 |- ( A i^i B ) e. _V
5		eleq1 2295	. . 3 |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) )
6	4, 5	mpbii 148	. 2 |- ( ( A i^i B ) = A -> A e. _V )
7	1, 6	sylbi 121	1 |- ( A C_ B -> A e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   C_ wss 3211  BOUNDED wbdc 16610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bdsep 16654

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-bdc 16611

This theorem is referenced by:  bdssexi  16673  bdssexg  16674  bdfind  16716