Theorem ssindif0im 3453

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 3345	. . 3 |- B C_ ( _V \ ( _V \ B ) )
2		sstr 3136	. . 3 |- ( ( A C_ B /\ B C_ ( _V \ ( _V \ B ) ) ) -> A C_ ( _V \ ( _V \ B ) ) )
3	1, 2	mpan2 422	. 2 |- ( A C_ B -> A C_ ( _V \ ( _V \ B ) ) )
4		disj2 3449	. 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 1335   _Vcvv 2712   \ cdif 3099   i^i cin 3101   C_ wss 3102   (/)c0 3394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by: (None)