Theorem bdssex 16037

Description: Bounded version of ssex 4197. (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 3187	. 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 16035	. . 3 |- ( A i^i B ) e. _V
5		eleq1 2270	. . 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 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   C_ wss 3174  BOUNDED wbdc 15975

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-bdc 15976

This theorem is referenced by:  bdssexi  16038  bdssexg  16039  bdfind  16081