Theorem ssini 3345

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 270	. 2 |- ( A C_ B /\ A C_ C )
4		ssin 3344	. 2 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )
5	3, 4	mpbi 144	1 |- A C_ ( B i^i C )

Syntax hints:   /\ wa 103   i^i cin 3115   C_ wss 3116

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

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

This theorem is referenced by:  inv1  3445