Theorem inss 3337

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 3336	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
2		incom 3299	. . 3 |- ( A i^i B ) = ( B i^i A )
3		ssinss1 3336	. . 3 |- ( B C_ C -> ( B i^i A ) C_ C )
4	2, 3	eqsstrid 3174	. 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 3101   C_ wss 3102

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115

This theorem is referenced by: (None)