Theorem inss 3393

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 3392	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
2		incom 3355	. . 3 |- ( A i^i B ) = ( B i^i A )
3		ssinss1 3392	. . 3 |- ( B C_ C -> ( B i^i A ) C_ C )
4	2, 3	eqsstrid 3229	. 2 |- ( B C_ C -> ( A i^i B ) C_ C )
5	1, 4	jaoi 717	1 |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   \/ wo 709   i^i cin 3156   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170

This theorem is referenced by: (None)