Theorem ssindif0im 3506

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 3397	. . 3 |- B C_ ( _V \ ( _V \ B ) )
2		sstr 3187	. . 3 |- ( ( A C_ B /\ B C_ ( _V \ ( _V \ B ) ) ) -> A C_ ( _V \ ( _V \ B ) ) )
3	1, 2	mpan2 425	. 2 |- ( A C_ B -> A C_ ( _V \ ( _V \ B ) ) )
4		disj2 3502	. 2 |- ( ( A i^i ( _V \ B ) ) = (/) <-> A C_ ( _V \ ( _V \ B ) ) )
5	3, 4	sylibr 134	1 |- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2760   \ cdif 3150   i^i cin 3152   C_ wss 3153   (/)c0 3446

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by: (None)