Theorem ssini 3400

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 3399	. 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 3169   C_ wss 3170

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 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183

This theorem is referenced by:  inv1  3501