Theorem ssinss1 3460

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 3451	. 2 |- ( A i^i B ) C_ A
2		sstr2 3255	. 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 3219   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  inss  3461  ballotfilemfrc  13248  insubm  13769  distop  15109  ntrin  15148  innei  15187  txcnp  15295