Theorem inss 3166

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 3165	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
2		incom 3129	. . 3 |- ( A i^i B ) = ( B i^i A )
3		ssinss1 3165	. . 3 |- ( B C_ C -> ( B i^i A ) C_ C )
4	2, 3	syl5eqss 2989	. 2 |- ( B C_ C -> ( A i^i B ) C_ C )
5	1, 4	jaoi 636	1 |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   \/ wo 629   i^i cin 2916   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by: (None)