Theorem bdssex 13488

Description: Bounded version of ssex 4101. (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 3115	. 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 13486	. . 3 |- ( A i^i B ) e. _V
5		eleq1 2220	. . 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 1335   e. wcel 2128   _Vcvv 2712   i^i cin 3101   C_ wss 3102  BOUNDED wbdc 13426

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bdsep 13470

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-bdc 13427

This theorem is referenced by:  bdssexi  13489  bdssexg  13490  bdfind  13532