Theorem ssini 3350

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 3349	. 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 3120   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134

This theorem is referenced by:  inv1  3451