Theorem ssinss1 3402

Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Assertion

Ref	Expression
ssinss1	|- ( A C_ C -> ( A i^i B ) C_ C )

Proof of Theorem ssinss1

Step	Hyp	Ref	Expression
1		inss1 3393	. 2 |- ( A i^i B ) C_ A
2		sstr2 3200	. 2 |- ( ( A i^i B ) C_ A -> ( A C_ C -> ( A i^i B ) C_ C ) )
3	1, 2	ax-mp 5	1 |- ( A C_ C -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   i^i cin 3165   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179

This theorem is referenced by:  inss  3403  insubm  13350  distop  14590  ntrin  14629  innei  14668  txcnp  14776