Theorem ssini 3360

Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Hypotheses

Ref	Expression
ssini.1	|- A C_ B
ssini.2	|- A C_ C

Assertion

Ref	Expression
ssini	|- A C_ ( B i^i C )

Proof of Theorem ssini

Step	Hyp	Ref	Expression
1		ssini.1	. . 3 |- A C_ B
2		ssini.2	. . 3 |- A C_ C
3	1, 2	pm3.2i 272	. 2 |- ( A C_ B /\ A C_ C )
4		ssin 3359	. 2 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )
5	3, 4	mpbi 145	1 |- A C_ ( B i^i C )

Syntax hints:   /\ wa 104   i^i cin 3130   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144

This theorem is referenced by:  inv1  3461