Theorem ssini 3395

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 3394	. 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 3164   C_ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178

This theorem is referenced by:  inv1  3496