Theorem ssinss1 3436

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 3427	. 2 |- ( A i^i B ) C_ A
2		sstr2 3234	. 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 3199   C_ wss 3200

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  inss  3437  insubm  13567  distop  14808  ntrin  14847  innei  14886  txcnp  14994