Theorem inss 3352

Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

Assertion

Ref	Expression
inss	|- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Proof of Theorem inss

Step	Hyp	Ref	Expression
1		ssinss1 3351	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
2		incom 3314	. . 3 |- ( A i^i B ) = ( B i^i A )
3		ssinss1 3351	. . 3 |- ( B C_ C -> ( B i^i A ) C_ C )
4	2, 3	eqsstrid 3188	. 2 |- ( B C_ C -> ( A i^i B ) C_ C )
5	1, 4	jaoi 706	1 |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   \/ wo 698   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: (None)