Theorem ssindif0im 3348

Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0im	|- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )

Proof of Theorem ssindif0im

Step	Hyp	Ref	Expression
1		ddifss 3240	. . 3 |- B C_ ( _V \ ( _V \ B ) )
2		sstr 3036	. . 3 |- ( ( A C_ B /\ B C_ ( _V \ ( _V \ B ) ) ) -> A C_ ( _V \ ( _V \ B ) ) )
3	1, 2	mpan2 417	. 2 |- ( A C_ B -> A C_ ( _V \ ( _V \ B ) ) )
4		disj2 3344	. 2 |- ( ( A i^i ( _V \ B ) ) = (/) <-> A C_ ( _V \ ( _V \ B ) ) )
5	3, 4	sylibr 133	1 |- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1290   _Vcvv 2622   \ cdif 2999   i^i cin 3001   C_ wss 3002   (/)c0 3289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-dif 3004  df-in 3008  df-ss 3015  df-nul 3290

This theorem is referenced by: (None)